Problem 1

Linear, constant, and squaring functions are examples of ______________ functions.

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Problem 1

An exponential growth model has the form__________and an exponential decay model has the form________

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Problem 2

A polynomial function of $x$ with degree $n$ has the form $$f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ $\left(a_{n} \neq 0\right),$ where $n$ is a_______and $a_{n+} a_{n-1}, . . . ., a_{1}, a_{0}$ are__________numbers.

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Problem 3

$\mathrm{A}$_________function is a second-degree polynomial function, and its graph is called a_______

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Problem 3

In probability and statistics, Gaussian models commonly represent populations that are__________

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Problem 5

When the graph of a quadratic function opens upward, its leading coefficient is_________and the vertex

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Problem 5

Solving for a variable In Exercises 5 and $6,(a)$ solve

for $P$ and $(b)$ solve fort.

$$A=P e^{r t}$$

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Problem 6

When the graph of a quadratic function opens downward, its leading coefficient is

vertex of the graph is a________and the vertex of the graph is a________

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Problem 6

Solving for a variable In Exercises 5 and $6,(a)$ solve

for $P$ and $(b)$ solve fort.

$$A=P\left(1+\frac{r}{n}\right)^{n t}$$

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Problem 7

Matching In Exercises $7-12,$ match the quadratic

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$f(x)=(x-2)^{2}$$

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Problem 7

Compound Interest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 8

Matching In Exercises $7-12,$ match the quadratic

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$f(x)=(x+4)^{2}$$

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Problem 8

Compound Interest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 9

Matching In Exercises $7-12,$ match the quadratic

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$

f(x)=x^{2}-2

$$

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Problem 9

Compound Interest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 10

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$

f(x)=(x+1)^{2}-2

$$

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Problem 10

Compound Interest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 11

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$

f(x)=4-(x-2)^{2}

$$

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Problem 11

Compound lnterest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 12

function with its graph. [The graphs are labeled (a), (b),

$(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),$ and $(\mathbf{f}) . ]$

$$

f(x)=-(x-4)^{2}

$$

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Problem 12

Compound lnterest In Exercises $7-12,$ complete the

table assuming continuously compounded interest.

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Problem 13

Sketching Graphs of Quadratic Functions In }} \\ {\text { Exercises } 13-16, \text { sketch the graph of each quadratic }} \\ {\text { function and compare it with the graph of } y=x^{2} .}\end{array}

$$

\begin{array}{ll}{\text { (a) } f(x)=\frac{1}{2} x^{2}} & {\text { (b) } g(x)=-\frac{1}{8} x^{2}} \\ {\text { (c) } h(x)=\frac{3}{2} x^{2}} & {\text { (d) } k(x)=-3 x^{2}}\end{array}

$$

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Problem 13

Compound Interest In Exercises 13 and $14,$

determine the principal $P$ that must be invested at rate $r$

compounded monthly, so that $\$ 500,000$ will be available

for retirement in $t$ years.

$$r=5 \%, t=10$$

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Problem 14

Sketching Graphs of Quadratic Functions In }} \\ {\text { Exercises } 13-16, \text { sketch the graph of each quadratic }} \\ {\text { function and compare it with the graph of } y=x^{2} .}\end{array}

$$

\begin{array}{ll}{\text { (a) } f(x)=x^{2}+1} & {\text { (b) } g(x)=x^{2}-1} \\ {\text { (c) } h(x)=x^{2}+3} & {\text { (d) } k(x)=x^{2}-3}\end{array}

$$

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Problem 14

Compound Interest In Exercises 13 and $14,$

determine the principal $P$ that must be invested at rate $r$

compounded monthly, so that $\$ 500,000$ will be available

for retirement in $t$ years.

$r=3 \frac{1}{2} \%, t=15$

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Problem 15

Sketching Graphs of Quadratic Functions In }} \\ {\text { Exercises } 13-16, \text { sketch the graph of each quadratic }} \\ {\text { function and compare it with the graph of } y=x^{2} .}\end{array}

$$

\begin{array}{ll}{\text { (a) } f(x)=(x-1)^{2}} & {\text { (b) } g(x)=(3 x)^{2}+1} \\ {\text { (c) } h(x)=\left(\frac{1}{3} x\right)^{2}-3} & {\text { (d) } k(x)=(x+3)^{2}}\end{array}

$$

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Problem 15

Compound Interest In Exercises 15 and $16,$

determine the time necessary for $P$ dollars to double

when it is invested at interest rate $r$ compounded

(a) annually, (b) monthly, (c) daily, and (d) continuously.

$$r=10 \%$$

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Problem 16

$$

\begin{array}{l}{\text { (a) } f(x)=-\frac{1}{2}(x-2)^{2}+1} \\ {\text { (b) } g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3} \\ {\text { (c) } h(x)=-\frac{1}{2}(x+2)^{2}-1} \\ {\text { (d) } k(x)=[2(x+1)]^{2}+4}\end{array}

$$

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Problem 16

Compound Interest In Exercises 15 and $16,$

determine the time necessary for $P$ dollars to double

when it is invested at interest rate $r$ compounded

(a) annually, (b) monthly, (c) daily, and (d) continuously.

$$r=6.5 \%$$

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Problem 17

Using Standard Form to Graph a Parabola In

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}-6 x

$$

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Problem 17

Compound lnterest Complete the table for the time

$t($ in years) necessary for $P$ dollars to triple when interest

is compounded (a) continuously and (b) annually at rate $r .$

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Problem 18

Using Standard Form to Graph a Parabola In

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

g(x)=x^{2}-8 x

$$

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Problem 18

Modeling Data Draw scatter plots of the data in

Exercise $17 .$ Use the regression feature of a graphing

utility to find models for the data.

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Problem 19

Using Standard Form to Graph a Parabola In

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

h(x)=x^{2}-8 x+16

$$

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Problem 19

Comparing Models If $\$ 1$ is invested over a

10 -year period, then the balance $A,$ where $t$ represents

the time in years, is given by $A=1+0.075[t]$ or

$A=e^{0.07 t}$ depending on whether the interest is simple

interest at 7$\frac{1}{2} \%$ or continuous compound interest at 7$\% .$ Graph each function on the same set of axes. Which

grows at a greater rate? (Remember that $[t]$ is the

greatest integer function discussed in Section $1.6 . )$

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Problem 20

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

g(x)=x^{2}+2 x+1

$$

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Problem 20

Comparing Models If $\$ 1$ is invested over a

10 -year period, then the balance $A,$ where $t$ represents

the time in years, is given by $A=1+0.06\|t\|$ or

$A=[1+(0.055 / 365)]^{1365 t ]}$ depending on whether the

interest is simple interest at 6$\%$ or compound interest at at

$5^{1} \%_{b}$ compounded daily. Use a graphing utility to graph

each function in the same viewing window. Which

grows at a greater rate?

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Problem 21

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}+8 x+13

$$

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Problem 21

Radioactive Decay In Exercises $21-24,$ complete the

table for the radioactive isotope.

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Problem 22

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}-12 x+44

$$

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Problem 22

Radioactive Decay In Exercises $21-24,$ complete the

table for the radioactive isotope.

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Problem 23

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}-14 x+54

$$

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Problem 23

Radioactive Decay In Exercises $21-24,$ complete the

table for the radioactive isotope.

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Problem 24

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

h(x)=x^{2}+16 x-17

$$

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Problem 24

Radioactive Decay In Exercises $21-24,$ complete the

table for the radioactive isotope.

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Problem 25

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}+34 x+289

$$

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Problem 25

$25-28,$ find the exponential model that fits the points

shown in the graph or table. as $y$

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Problem 26

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}-30 x+225

$$

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Problem 26

$25-28,$ find the exponential model that fits the points

shown in the graph or table. as $y$

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Problem 27

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}-x+\frac{5}{4}

$$

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Problem 27

$25-28,$ find the exponential model that fits the points

shown in the graph or table. as $y$

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Problem 28

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=x^{2}+3 x+\frac{1}{4}

$$

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Problem 28

$25-28,$ find the exponential model that fits the points

shown in the graph or table. as $y$

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Problem 29

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=-x^{2}+2 x+5

$$

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Problem 29

Population The populations $P$ (in thousands) of

Horry County, South Carolina, from 1980 through 2010

can be modeled by

$$P=20.6+85.5 e^{0.0360 t}$$

where $t$ represents the year, with $t=0$ corresponding to

$1980 .$ (Source: U.S. Census Bureau)

(a) Use the model to complete the table.

(b) According to the model, when will the population

of Horry County reach $350,000 ?$

(c) Do you think the model is valid for long-term

predictions of the population? Explain.

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Problem 30

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=-x^{2}-4 x+1

$$

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Problem 30

The table shows the mid-year populations (in millions)

of five countries in 2010 and the projected populations(in millions) for the year $2020 . \quad$ Source: U.S.

Census Bureau)

(a) Find the exponential growth or decay model

$y=a e^{b t}$ or $y=a e^{-b t}$ for the population of each

country by letting $t=10$ correspond to $2010 .$ Use

the model to predict the population of each country

in 2030

You can see that

the populations of

the United States

and the United

Kingdom are growing at

different rates.

What constant in

the equation

$y=a e^{b t}$ gives the growth rate? Discuss the

relationship between the different growth rates

and the magnitude of the constant.

(c) You can see that the population of China is

increasing, whereas the population of Bulgaria

is decreasing. What constant in the equation

$y=a e^{b t}$ reflects this difference? Explain.

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Problem 31

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

h(x)=4 x^{2}-4 x+21

$$

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Problem 31

Website Growth The number $y$ of hits a new

website receives each month can be modeled by

$y=4080 e^{k t},$ where $t$ represents the number of months

the website has been operating. In the website's thirdmonth, there were $10,000$ hits. Find the value of $k,$ and

use this value to predict the number of hits the website

will receive after 24 months.

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Problem 32

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=2 x^{2}-x+1

$$

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Problem 32

Population The populations $P$ (in thousands) of

Tallahassee, Florida, from 2005 through 2010 can be

modeled by $P=319.2 e^{k t},$ where $t$ represents the

year, with $t=5$ corresponding to $2005 .$ In $2006,$ the population of Tallahassee was about $347,000 .$

(Source: U.S. Census Bureau)

(a) Find the value of $k .$ Is the population increasing or

decreasing? Explain.

(b) Find the exponential model $V=a e^{k t}$

(c) Use a graphing utility to graph the two models in the

same viewing window. Which model depreciates

faster in the first 2 years?

(d) Find the book values of the computer after 1 year

and after 3 years using each model.

(e) Explain the advantages and disadvantages of using

each model to a buyer and a seller.

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Problem 33

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=\frac{1}{4} x^{2}-2 x-12

$$

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Problem 33

Bacteria Growth The number of bacteria in a

culture is increasing according to the law of exponential

growth. The initial population is 250 bacteria, and the

population after 10 hours is double the population after

1 hour. How many bacteria will there be after 6 hours?

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Problem 34

Exercises $17-34$ , write the quadratic function in

standard form and sketch its graph. Identify the vertex,

axis of symmetry, and $x$ -intercept(s).

$$

f(x)=-\frac{1}{3} x^{2}+3 x-6

$$

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Problem 34

culture is increasing according to the law of exponential

growth. The initial population is 250 bacteria, and the

population after 10 hours is double the population after

1 hour. How many bacteria will there be after 6 hours?

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Problem 35

Graphical Analysis In Exercises $35-42,$ use a

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=-\left(x^{2}+2 x-3\right)

$$

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Problem 35

Depreciation A laptop computer that costs $\$ 1150$

new has a book value of $\$ 550$ after 2 years.

(a) Find the linear model $V=m t+b .$

(b) Find the exponential model $V=a e^{k t} .$

(c) Use a graphing utility to graph the two models in the

same viewing window. Which model depreciates

faster in the first 2 years?

(d) Find the book values of the computer after 1 year

and after 3 years using each model.

(e) Explain the advantages and disadvantages of using

each model to a buyer and a seller.

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Problem 36

Graphical Analysis In Exercises $35-42,$ use a

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=-\left(x^{2}+x-30\right)

$$

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Problem 36

Learning Curve The management at a plastics

factory has found that the maximum number of units

a worker can produce in a day is $30 .$ The learning curve

for the number $N$ of units produced per day after a

new employee has worked t days is modeled by

$N=30\left(1-e^{k t}\right) .$ After 20 days on the job, a new

employee produces 19 units.

(a) Find the learning curve for this employee (first, find

the value of $k )$

(b) How many days should pass before this employee

is producing 25 units per day?

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Problem 37

Graphical Analysis In Exercises $35-42,$ use a

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

g(x)=x^{2}+8 x+11

$$

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Problem 37

Carbon Dating

(a) The ratio of carbon 14 to carbon 12 in a piece of

wood discovered in a cave is $R=1 / 8^{44} .$ Estimate

the age of the piece of wood.

(b) The ratio of carbon 14 to carbon 12 in a piece of

paper buried in a tomb is $R=1 / 13^{11} .$ Estimate the

age of the piece of paper.

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Problem 38

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=x^{2}+10 x+14

$$

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Problem 38

Radioactive Decay Carbon 14 dating assumes

that the carbon dioxide on Earth today has the same

radioactive content as it did centuries ago. If this is true,

then the amount of $^{14} \mathrm{C}$ absorbed by a tree that grew

several centuries ago should be the same as the amount

of 14 C absorbed by a tree growing today. A piece of

ancient charcoal contains only 15$\%$ as much radioactive

carbon as a piece of modern charcoal. How long ago

was the tree burned to make the ancient charcoal,

assuming that the half-life of $^{14} \mathrm{C}$ is 5715 years?

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Problem 39

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=2 x^{2}-16 x+32

$$

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Problem 39

IQ Scores The IQ scores for a sample of a class of

returning adult students at a small northeastern college

roughly follow the normal distribution

$$y=0.0266 e^{-(x-100)^{2} / 450}, \quad 70 \leq x \leq 115$$

where $x$ is the IQ score.

(a) Use a graphing utility to graph the function.

(b) From the graph in part (a), estimate the average

number of hours per week a student uses the

tutoring center.

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Problem 39

IQ Scores The IQ scores for a sample of a class of

returning adult students at a small northeastern college

roughly follow the normal distribution

$$y=0.0266 e^{-(x-100)^{2} / 450}, \quad 70 \leq x \leq 115$$

where $x$ is the IQ score.

(a) Use a graphing utility to graph the function.

(b) From the graph in part (a), estimate the average

IQ score of an adult student.

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Problem 40

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=-4 x^{2}+24 x-41

$$

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Problem 40

Education The amount of time (in hours per week)

a student utilizes a math-tutoring center roughly follows

the normal distribution

$$y=0.7979 e^{-(x-5.4)^{2} / 0.5,}, 4 \leq x \leq 7$$

where $x$ is the number of hours.

(a) Use a graphing utility to graph the function.

(b) From the graph in part (a), estimate the averag.

number of hours per week a student uses the

tutoring center.

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Problem 41

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

g(x)=\frac{1}{2}\left(x^{2}+4 x-2\right)

$$

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Problem 41

Cell Sites A cell site is a site where electronic

communications equipment is placed in a cellular

network for the use of mobile phones. The numbers y of

cell sites from 1985 through 2011 can be modeled by

$$y=\frac{269,573}{1+985 e^{-0.308 t}}$$

where $t$ represents the year, with $t=5$ corresponding to

$1985 . \quad$ (Source: CTIA-The Wireless Association)

(a) Use the model to find the numbers of cell sites in

the years $1998,2003,$ and $2006 .$

(b) Use a graphing utility to graph the function.

(c) Use the graph to determine the year in which the

number of cell sites reached $250,000 .$

(d) Confirm your answer to part (c) algebraically.

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Problem 42

graphing utility to graph the quadratic function.

Identify the vertex, axis of symmetry, and $x$ -intercept(s).

Then check your results algebraically by writing the

quadratic function in standard form.

$$

f(x)=\frac{3}{5}\left(x^{2}+6 x-5\right)

$$

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Problem 42

Population The populations $P$ (in thousands) of a

city from 2000 through 2010 can be modeled by

$$P=\frac{2632}{1+0.083 e^{0.050}}$$

where $t$ represents the year, with $t=0$ corresponding to

$2000 .$

(a) Use the model to find the populations of the city in

the years $2000,2005,$ and $2010 .$

(b) Use a graphing utility to graph the function.

(c) Use the graph to determine the year in which the

population will reach $2,2$ million.

(d) Confirm your answer to part (c) algebraically.

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Problem 43

Writing the Equation of a Parabola In Exercises

$43-46,$ write an equation for the parabola in standard

form.

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Problem 43

Population Growth A conservation organization

released 100 animals of an endangered species into a

game preserve. The preserve has a carrying capacity of

1000 animals. The growth of the pack is modeled by the

logistic curve

$$p(t)=\frac{1000}{1+9 e^{-0.1656 t}}$$

where $t$ is measured in months (see figure).

(a) Estimate the population after 5 months.

(b) After how many months is the population 500$?$

(c) Use a graphing utility to graph the function. Use thh

graph to determine the horizontal asymptotes, and

interpret the meaning of the asymptotes in the

context of the problem.

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Problem 44

$43-46,$ write an equation for the parabola in standard

form.

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Problem 44

Sales After discontinuing all advertising for a tool

kit in 2007 , the manufacturer noted that sales began to

drop according to the model

$$S=\frac{500,000}{1+0.4 e^{k t}}$$

where $S$ represents the number of units sold and $t=7$

represents $2007 .$ In $2011,300,000$ units were sold.

(a) Complete the model by solving for $k$ .

(b) Estimate sales in $2015 .$

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Problem 45

$43-46,$ write an equation for the parabola in standard

form.

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Problem 45

Find the intensity $I$ of an earthquake measuring $R$ on the

Richter scale (let $I_{0}=1 )$ .

(a) South Shetland Islands in $2012 : R=6.6$

(b) Oklahoma in $2011 : R=5.6$

(c) Papua New Guinea in $2011 \cdot R=71$

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Problem 46

$43-46,$ write an equation for the parabola in standard

form.

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Problem 46

Find the magnitude $R$ of each earthquake of intensity $I$

(a) $I=199,500,000$

(b) $I=48,275,000$

(c) $I=17,000$

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Problem 47

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(-2,5) ;$ point: $(0,9)$

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Problem 47

Intensity of Sound In Exercises $47-50$ , use the

following information for determining sound intensity.

The level of sound $\beta,$ in decibels, with an intensity

of $I,$ is given by $\beta=10 \log \left(I I_{0}\right),$ where $I_{0}$ is an

intensity of $10^{-12}$ watt per square meter, corresponding roughly to the faintest sound that can be heard by the

human ear. In Exercises 47 and $48,$ find the level of

sound $\beta .$

(a) $I=10^{-10}$ watt per $\mathrm{m}^{2}$ (quiet room)

(b) $I=10^{-5}$ watt per $\mathrm{m}^{2}$ (busy street corner)

(c) $I=10^{-8}$ watt per $\mathrm{m}^{2}$ (quiet radio)

(d) $I=10^{0}$ watt per $\mathrm{m}^{2}$ (threshold of pain)

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Problem 48

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(4,-1) ;$ point: $(2,3)$

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Problem 48

Intensity of Sound In Exercises $47-50$ , use the

following information for determining sound intensity.

The level of sound $\beta,$ in decibels, with an intensity

of $I,$ is given by $\beta=10 \log \left(I I_{0}\right),$ where $I_{0}$ is an

intensity of $10^{-12}$ watt per square meter, corresponding roughly to the faintest sound that can be heard by the

human ear. In Exercises 47 and $48,$ find the level of

sound $\beta .$

(a) $I=10^{-11}$ watt per $\mathrm{m}^{2}$ (rustle of leaves)

(b) $I=10^{2}$ watt per $\mathrm{m}^{2}$ (jet at 30 meters)

(c) $I=10^{-4}$ watt per $\mathrm{m}^{2}$ (door slamming)

(d) $I=10^{-2}$ watt per $\mathrm{m}^{2}$ (siren at 30 meters)

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Problem 49

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(1,-2) ;$ point: $(-1,14)$

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Problem 49

Intensity of Sound In Exercises $47-50$ , use the

following information for determining sound intensity.

The level of sound $\beta,$ in decibels, with an intensity

of $I,$ is given by $\beta=10 \log \left(I I_{0}\right),$ where $I_{0}$ is an

intensity of $10^{-12}$ watt per square meter, corresponding roughly to the faintest sound that can be heard by the

human ear. In Exercises 47 and $48,$ find the level of

sound $\beta .$

Due to the installation of noise suppression materials,

the noise level in an auditorium decreased from 93 to

80 decibels. Find the percent decrease in the intensity

level of the noise as a result of the installation of these

materials.

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Problem 50

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(2,3) ;$ point: $(0,2)$

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Problem 50

following information for determining sound intensity.

The level of sound $\beta,$ in decibels, with an intensity

of $I,$ is given by $\beta=10 \log \left(I I_{0}\right),$ where $I_{0}$ is an

intensity of $10^{-12}$ watt per square meter, corresponding roughly to the faintest sound that can be heard by the

human ear. In Exercises 47 and $48,$ find the level of

sound $\beta .$

Due to the installation of a muffler, the noise level of an

engine decreased from 88 to 72 decibels. Find the

percent decrease in the intensity level of the noise as

a result of the installation of the muffler.

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Problem 51

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(5,12) ;$ point: $(7,15)$

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Problem 51

pH Levels In Exercises $51-56$ , use the acidity model

given by $p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],$ where acidity $(\mathbf{p} \mathbf{H})$ is

a measure of the hydrogen ion concentration $\left[\mathbf{H}^{+}\right]$

(measured in moles of hydrogen per liter) of a solution.

Find the $\mathrm{pH}$ when $\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}$

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Problem 52

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(-2,-2) ;$ point: $(-1,0)$

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Problem 52

pH Levels In Exercises $51-56$ , use the acidity model

given by $p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],$ where acidity $(\mathbf{p} \mathbf{H})$ is

a measure of the hydrogen ion concentration $\left[\mathbf{H}^{+}\right]$

(measured in moles of hydrogen per liter) of a solution.

Find the pH when $\left[\mathrm{H}^{+}\right]=1.13 \times 10^{-5}$

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Problem 53

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $\left(-\frac{1}{4}, \frac{3}{2}\right) ;$ point: $(-2,0)$

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Problem 53

pH Levels In Exercises $51-56$ , use the acidity model

given by $p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],$ where acidity $(\mathbf{p} \mathbf{H})$ is

a measure of the hydrogen ion concentration $\left[\mathbf{H}^{+}\right]$

(measured in moles of hydrogen per liter) of a solution.

Compute $\left[\mathrm{H}^{+}\right]$ for a solution in which $\mathrm{pH}=5.8$

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Problem 54

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $\left(\frac{3}{2},-\frac{3}{4}\right) ;$ point: $(-2,4)$

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Problem 54

given by $p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],$ where acidity $(\mathbf{p} \mathbf{H})$ is

a measure of the hydrogen ion concentration $\left[\mathbf{H}^{+}\right]$

(measured in moles of hydrogen per liter) of a solution.

Compute $\left[\mathrm{H}^{+}\right]$ for a solution in which $\mathrm{pH}=3.2$

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Problem 55

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $\left(-\frac{5}{2}, 0\right) ;$ point: $\left(-\frac{7}{2},-\frac{16}{3}\right)$

Check back soon!

Problem 55

Apple juice has a pH of 2.9 and drinking water has a pH

of $8.0 .$ The hydrogen ion concentration of the apple

juice is how many times the concentration of drinking

water?

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Problem 56

Writing the Equation of a Parabola In Exercises

$47-56$ , write the standard form of the equation of the

parabola that has the indicated vertex and passes

through the given point.

Vertex: $(6,6) ;$ point: $\left(\frac{61}{10}, \frac{3}{2}\right)$

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Problem 56

The pH of a solution decreases by one unit. By what

factor does the hydrogen ion concentration increase?

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Problem 57

Graphical Reasoning In Exercises 57 and 58 ,

determine the $x$ -intercept(s) of the graph visually. Then

find the $x$ -intercept(s) algebraically to confirm your

results.

$$y=x^{2}-4 x-5$$

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Problem 57

Forensics At $8 : 30$ A.M., a coroner went to the

home of a person who had died during the night.

In order to estimate the time of death, the coroner

took the person's temperature twice. At $9 : 00$ A.M. the

temperature was $85.7^{\circ} \mathrm{F},$ and at $11 : 00$ A.M. thetemperature was $82.8^{\circ} \mathrm{F}$ . From these two temperatures,

the coroner was able to determine that the time elapsed

since death and the body temperature were related by

the formula

$$t=-10 \ln \frac{T-70}{98.6-70}$$

where $t$ is the time in hours elapsed since the person

died and $T$ is the temperature (in degrees Fahrenheit) of

the person's body. (This formula comes from a general

cooling principle called Newton's Law of Cooling.It uses the assumptions that the person had a normal

body temperature of $98.6^{\circ} \mathrm{F}$ at death and that the room

temperature was a constant $70^{\circ} \mathrm{F} .$ ) Use the formula to

estimate the time of death of the person.

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Problem 58

Graphical Reasoning In Exercises 57 and 58 ,

determine the $x$ -intercept(s) of the graph visually. Then

find the $x$ -intercept(s) algebraically to confirm your

results.

$$y=2 x^{2}+5 x-3$$

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Problem 58

Home Mortgage A $\$ 120,000$ home mortgage for

30 years at 7$\frac{1}{2} \%$ has a monthly payment of $\$ 839.06$

Part of the monthly payment covers the interest charge

on the unpaid balance, and the remainder of the

payment reduces the principal. The amount paid toward

the interest is

$$u=M-\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$

and the amount paid toward the reduction of the

principal is

$$v=\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$

In these formulas, $P$ is the size of the mortgage, $r$ is the

interest rate, $M$ is the monthly payment, and $t$ is the time

(in years).

(a) Use a graphing utility to graph each function

in the same viewing window. (The viewing

window should show all 30 years of mortgage

payments.)

(b) In the early years of the mortgage, is the greater part

of the monthly payment paid toward the interest or

the principal? Approximate the time when the

monthly payment is evenly divided between interest

and principal reduction.

(c) Repeat parts $(\mathrm{a})$ and (b) for a repayment period of

20 years $(M=\$ 966.71) .$ What can you conclude?

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Problem 59

Graphical Analysis In Exercises $59-64,$ use a

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=x^{2}-4 x

$$

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Problem 59

Home Mortgage The total interest $u$ paid on a home

mortgage of $P$ dollars at interest rate $r$ for $t$ years is

$$u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]$$

Consider a $\$ 120,000$ home mortgage at 7$\frac{1}{2} \%$

(a) Use a graphing utility to graph the total interest

function.

(b) Approximate the length of the mortgage for which

the total interest paid is the same as the size of the

mortgage. Is it possible that some people are paying

twice as much in interest charges as the size of the

mortgage?

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Problem 60

Graphical Analysis In Exercises $59-64,$ use a

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=-2 x^{2}+10 x

$$

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Problem 60

Data Analysis The table shows the time $t$ (in

seconds) required for a car to attain a speed of $s$ miles

per hour from a standing start.

Two models for these data are as follows.

$$t_{1}=40.757+0.556 s-15.817 \ln s$$

$$t_{2}=1.2259+0.0023 s^{2}$$

(a) Use the regression feature of a graphing utility to

find a linear model $t_{3}$ and an exponential model $t_{4}$

for the data.

(b) Use the graphing utility to graph the data and each

model in the same viewing window.

(c) Create a table comparing the data with estimates

obtained from each model

(d) Use the results of part (c) to find the sum of the

absolute values of the differences between the data

and the estimated values given by each model.

Based on the four sums, which model do you think

best fits the data? Explain.

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Problem 61

Graphical Analysis In Exercises $59-64,$ use a

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=x^{2}-9 x+18

$$

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Problem 61

True or False? In Exercises $61-64$ , determine

whether the statement is true or false. Justify your

answer.

The domain of a logistic growth function cannot be the

set of real numbers.

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Problem 62

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=x^{2}-8 x-20

$$

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Problem 62

True or False? In Exercises $61-64$ , determine

whether the statement is true or false. Justify your

answer.

A logistic growth function will always have an $x$ -intercept.

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Problem 63

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=2 x^{2}-7 x-30

$$

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Problem 63

True or False? In Exercises $61-64$ , determine

whether the statement is true or false. Justify your

answer.

The graph of $f(x)=\frac{4}{1+6 e^{-2 x}}+5$ is the graph of

$$g(x)=\frac{4}{1+6 e^{-2 x}}$$ shifted to the right five units.

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Problem 64

graphing utility to graph the quadratic function. Find

the $x$ -intercept(s) of the graph and compare them with

the solutions of the corresponding quadratic equation

when $f(x)=0 .$

$$

f(x)=\frac{7}{10}\left(x^{2}+12 x-45\right)

$$

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Problem 64

True or False? In Exercises $61-64$ , determine

whether the statement is true or false. Justify your

answer.

The graph of a Gaussian model will never have an

$x$ -intercept.

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Problem 65

Finding Quadratic Functions In Exercises $65-70$ ,

find two quadratic functions, one that opens upward and

one that opens downward, whose graphs have the given

$x$ -intercepts. (There are many correct answers.)

$$

(-1,0),(3,0)

$$

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Problem 65

Writing Use your school's library, the Internet, or

some other reference source to write a paper describing

John Napier's work with logarithms.

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Problem 66

Finding Quadratic Functions In Exercises $65-70$ ,

find two quadratic functions, one that opens upward and

one that opens downward, whose graphs have the given

$x$ -intercepts. (There are many correct answers.)

$$

(-5,0),(5,0)

$$

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Problem 66

HOW DO YOU SEE IT? Identify each

model as exponential growth, exponential

decay, Gaussian, linear, logarithmic, logistic

growth, quadratic, or none of the above.

Explain your reasoning.

Project: Sales per Share To work an extended

application analyzing the sales per share for Kohl's

Corporation from 1995 through $2010,$ visit this text's website

at LarsonPrecalculus.com. (Source: Kohl's Corporation)

Check back soon!

Problem 67

Finding Quadratic Functions In Exercises $65-70$ ,

find two quadratic functions, one that opens upward and

one that opens downward, whose graphs have the given

$x$ -intercepts. (There are many correct answers.)

$$

(0,0),(10,0)

$$

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Problem 68

find two quadratic functions, one that opens upward and

one that opens downward, whose graphs have the given

$x$ -intercepts. (There are many correct answers.)

$$

(4,0),(8,0)

$$

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Problem 69

find two quadratic functions, one that opens upward and

one that opens downward, whose graphs have the given

$x$ -intercepts. (There are many correct answers.)

$$

(-3,0),\left(-\frac{1}{2}, 0\right)

$$

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Problem 71

Number Problems In Exercises $71-74,$ find two

positive real numbers whose product is a maximum.

The sum is $110 .$

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Problem 72

Number Problems In Exercises $71-74,$ find two

positive real numbers whose product is a maximum.

The sum is $S$

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Problem 73

Number Problems In Exercises $71-74,$ find two

positive real numbers whose product is a maximum.

The sum of the first and twice the second is $24 .$

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Problem 74

Number Problems In Exercises $71-74,$ find two

positive real numbers whose product is a maximum.

The sum of the first and three times the second is 42

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Problem 75

Path of a Diver

$$f(x)=-\frac{4}{9} x^{2}+\frac{24}{9} x+12$$

where $f(x)$ is the

height (in feet) and

$x$ is the horizontal

distance from the end

of the diving board

(in feet). What is the

maximum height of

the diver?

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Problem 76

Height of a Ball The path of a punted football is

given by the function

$$f(x)=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5$$

where $f(x)$ is the height (in feet) and $x$ is the horizontalal

distance (in feet) from the point at which the ball is

punted.

(a) How high is the ball when it is punted?

(b) What is the maximum height of the punt?

(c) How long is the punt?

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Problem 77

Minimum cost A manufacturer of lighting fixtures

has daily production costs of $C=800-10 x+0.25 x^{2}$ ,

where $C$ is the total cost (in dollars) and $x$ is the

number of units produced. How many fixtures should

be produced each day to yield a minimum cost?

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Problem 78

Maximum Profit The profit $P$ (in hundreds

of dollarss that a company makes depends on the ehe

amount $x$ (in hundreds of dollars) the company y

spends on advertising according to the model $\vec{P}=230+20 x-0.5 x^{2} . \quad$ What expenditure for

advertising will yield a maximum profit?

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Problem 79

Maximum Revenue The total revenue $R$ earned

(in thousands of dollars) from manufacturing handheld

video games is given by

$$R(n)=-25 n^{2}+1200 n$$

where $p$ is the price per unit (in dollars).

(a) Find the revenues when the prices per unit are $\$ 20,$

$\$ 25,$ and $\$ 30 .$

(b) Find the unit price that will yield a maximum

revenue. What is the maximum revenue? Explain

your results.

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Problem 80

Maximum Revenue The total revenue $R$ earned

per day (in dollars) from a pet-sitting service is given by

$R(p)=-12 p^{2}+150 p,$ where $p$ is the price charged

per pet (in dollars).

(a) Find the revenues when the prices per pet are $\$ 4$

$\$ 6,$ and $\$ 8 .$

(b) Find the unit price that will yield a maximum

revenue. What is the maximum revenue? Explain

your results.

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Problem 81

Numerical, Graphical, and Analytical Analysis

A rancher has 200 feet of fencing to enclose two

adjacent rectangular corrals (see figure).

(a) Write the area $A$ of the corrals as a function of $x .$

(b) Construct a table showing possible values of $x$ and

the corresponding areas of the corral. Use the table

to estimate the dimensions that will produce the

maximum enclosed area.

(c) Use a graphing utility to graph the area function.

Use the graph to approximate the dimensions that

will produce the maximum enclosed area.

(d) Write the area function in standard form to find

analytically the dimensions that will produce the

maximum area

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Problem 82

Geometry An indoor physical fitness room consists

of a rectangular region with a semicircle on each end.The perimeter of the room is to be a 200 -meter

single-lane running track.

(a) Draw a diagram that gives a visual representation of

the problem. Let $x$ and $y$ represent the length and

width of the rectangular region, respectively.

(b) Determine the radius of each semicircular end of the

room. Determine the distance, in terms of $y,$ around

the inside edge of each semicircular part of the track.

(c) Use the result of part (b) to write an equation, in

terms of $x$ and $y,$ for the distance traveled in one lap

around the track. Solve for $y .$

(d) Use the result of part (c) to write the area $A$ of

the rectangular region as a function of $x .$ What

dimensions will produce a rectangle of maximum

area?

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Problem 83

Maximum Revenue A small theater has a seating

capacity of $2000 .$ When the ticket price is $\$ 20,$

attendance is $1500 .$ For each $\$ 1$ decrease in price,

attendance increases by 100 .

(a) Write the revenue $R$ of the theater as a function of

ticket price $x .$

(b) What ticket price will yield a maximum revenue?

What is the maximum revenue?

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Problem 84

Maximum Area A Norman window is constructed

by adjoining a semicircle to the top of an ordinary

rectangular window (see figure). The perimeter of the

window is 16 feet.

(a) Write the area $A$ of the window as a function of $x$ .

(b) What dimensions will produce a window of

maximum area?

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Problem 85

Graphical Analysis From 1950 through $2005,$ the

per capita consumption $C$ of cigarettes by Americans

(age 18 and older) can be modeled by

$C=3565.0+60.30 t-1.783 t^{2}, 0 \leq t \leq 55,$ where $t$

is the year, with $t=0$ corresponding to $1950 .$

(Source: Tobacco Outlook Report)

(a) Use a graphing utility to graph the model.

(b) Use the graph of the model to approximate the

maximum average annual consumption. Beginning

in $1966,$ all cigarette packages were required by law

to carry a health warning. Do you think the warning

had any effect? Explain.

(c) In $2005,$ the U.S. population (age 18 and over) was

$296,329,000 .$ Of those, about $59,858,458$ were

smokers. What was the average annual cigarette

consumption per smoker in 2005$?$ What was the

average daily cigarette consumption per smoker?

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Problem 86

Data Analysis: Sales The sales $y$ (in billions of

dollarss for Harley-Davidson from 2000 through 2010 are

shown in the table. (Source: U.S. Harley-Davidson, Inc.)

(a) Use a graphing utility to create a scatter plot of

the data. Let $x$ represent the year, with $x=0$

corresponding to $2000 .$

(b) Use the regression feature of the graphing utility to

find a quadratic model for the data.

(c) Use the graphing utility to graph the model in the

same viewing window as the scatter plot. How wel

does the model fit the data?

(d) Use the trace feature of the graphing utility to

approximate the year in which the sales for

Harley-Davidson were the greatest.

(e) Verify your answer to part (d) algebraically.

(f) Use the model to predict the sales for Harley-Davidson

in $2013 .$

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Problem 87

True or False? In Exercises 87 and 88 , determine

whether the statement is true or false. Justify your

answer.

The graph of $f(x)=-12 x^{2}-1$ has no $x$ -intercepts.

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Problem 88

The graphs of

$$f(x)=-4 x^{2}-10 x+7$$

and

$$g(x)=12 x^{2}+30 x+1$$

have the same axis of symmetry.

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Problem 89

Think About It In Exercises $89-92$ , find the values of

$b$ such that the function has the given maximum or

minimum value.

$$f(x)=-x^{2}+b x-75 ;$ Maximum value: 25$

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Problem 90

Think About It In Exercises $89-92$ , find the values of

$b$ such that the function has the given maximum or

minimum value.

$f(x)=-x^{2}+b x-16 ;$ Maximum value: 48

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Problem 91

Think About It In Exercises $89-92$ , find the values of

$b$ such that the function has the given maximum or

minimum value.

$f(x)=x^{2}+b x+26 ;$ Minimum value: 10

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Problem 92

Think About It In Exercises $89-92$ , find the values of

$b$ such that the function has the given maximum or

minimum value.

$f(x)=x^{2}+b x-25 ;$ Minimum value: $-50$

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Problem 93

Verifying the Vertex Write the quadratic function

$$f(x)=a x^{2}+b x+c$$

in standard form to verify that the vertex occurs at

$$\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)$$

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Problem 94

HOW DO YOU SEE IT? The graph shows a

quadratic function of the form

$$P(t)=a t^{2}+b t+c$$

which represents the yearly profits for a

company, where $P(t)$ is the profit in year $t$ .

(a) Is the value of $a$ positive, negative, or zero?

Explain.

(b) Write an expression in terms of $a$ and $b$ that

represents the year $t$ when the company made

the least profit.

(c) The company made the same yearly profits in

2004 and $2012 .$ Estimate the year in which the

company made the least profit.

(d) Assume that the model is still valid today.

Are the yearly profits currently increasing,

decreasing, or constant? Explain.

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Problem 95

Proof Assume that the function

$$f(x)=a x^{2}+b x+c, \quad a \neq 0$$

has two real zeros. Prove that the $x$ -coordinate of the

vertex of the graph is the average of the zeros of $f$ .

(Hint: Use the Quadratic Formula.)

Project: Height of a Basketball To work ar

extended application analyzing the height of a basketbal

after it has been dropped, visit this text's website a

LarsonPrecalculus.com.

Jorge M.

Numerade Educator