# Precalculus with Limits

## Educators JK
FH
EB
NM
+ 3 more educators

### 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 2

A logarithmic model has the form___________Or_________

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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 4

A logistic growth model has the form_________

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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 $(\mathbf{f}) . ]$

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

### 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

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 $(\mathbf{f}) . ]$

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

JK
J'Nae K.

### Problem 10

Compound Interest In Exercises $7-12,$ complete the
table assuming continuously compounded interest.

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### Problem 11

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 $(\mathbf{f}) . ]$

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

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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

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 $(\mathbf{f}) . ]$

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

FH
Faith H.

### 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. $$r=5 \%, t=10$$ Check back soon! ### 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} $$\begin{array}{ll}{\text { (a) } f(x)=x^{2}+1} & {\text { (b) } g(x)=x^{2}-1} \\ {\text { (c) } h(x)=x^{2}+3} & {\text { (d) } k(x)=x^{2}-3}\end{array}$$ EB Emma B. Numerade Educator ### 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.

$$r=10 \%$$ Deboney H.

### Problem 16

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}

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

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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).

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

NM
Nicholas M.

### 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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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$1.6 . )$Check back soon! ### Problem 20 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). $$g(x)=x^{2}+2 x+1$$ Check back soon! ### 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

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).

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

SC
Shonte C.

### Problem 21

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

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### Problem 22

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).

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

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### Problem 22

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

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### Problem 23

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).

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

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### Problem 23

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

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### Problem 24

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).

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

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### Problem 24

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

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### Problem 25

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).

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

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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

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).

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

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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

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).

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

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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

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).

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

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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

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).

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

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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

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).

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

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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

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).

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

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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

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### Problem 32

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).

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

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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.
each model to a buyer and a seller.

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### Problem 33

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).

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

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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

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).

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

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### Problem 34

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 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

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

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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.
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

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

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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

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

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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

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

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

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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

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

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

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

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

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

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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

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

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

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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 $250,000 .$

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### Problem 42

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

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

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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.

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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

Writing the Equation of a Parabola In Exercises
$43-46,$ 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

Writing the Equation of a Parabola In Exercises
$43-46,$ 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

Writing the Equation of a Parabola In Exercises
$43-46,$ 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

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 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

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 $\mathrm{pH}=3.2$

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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: $\left(-\frac{7}{2},-\frac{16}{3}\right)$

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### 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? Check back soon! ### 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 .$$$f(x)=x^{2}-4 x$$ Check back soon! ### 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
The domain of a logistic growth function cannot be the
set of real numbers.

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### Problem 62

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 $f(x)=0 .$

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

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### Problem 62

True or False? In Exercises $61-64$ , determine
whether the statement is true or false. Justify your
A logistic growth function will always have an $x$ -intercept.

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### Problem 63

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 $f(x)=0 .$

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

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### Problem 63

True or False? In Exercises $61-64$ , determine
whether the statement is true or false. Justify your
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

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 $f(x)=0 .$

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

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### Problem 64

True or False? In Exercises $61-64$ , determine
whether the statement is true or false. Justify your
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.
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)

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### 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

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.)

$$(4,0),(8,0)$$

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### Problem 69

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.)

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

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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. Check back soon! ### 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

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### Problem 81

Numerical, Graphical, and Analytical Analysis
A rancher has 200 feet of fencing to enclose two
(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
(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.
(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
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.
$$f(x)=-x^{2}+b x-75 ; Maximum value: 25 Check back soon! ### Problem 90 Think About It In Exercises 89-92 , find the values of b such that the function has the given maximum or minimum value. f(x)=-x^{2}+b x-16 ; Maximum value: 48 Check back soon! ### Problem 91 Think About It In Exercises 89-92 , find the values of b such that the function has the given maximum or minimum value. f(x)=x^{2}+b x+26 ; Minimum value: 10 Check back soon! ### Problem 92 Think About It In Exercises 89-92 , find the values of b such that the function has the given maximum or minimum value. f(x)=x^{2}+b x-25 ; Minimum value: -50 Check back soon! ### Problem 93 Verifying the Vertex Write the quadratic function$$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)$$Check back soon! ### Problem 94 HOW DO YOU SEE IT? The graph shows a quadratic function of the form$$P(t)=a t^{2}+b t+c$$which represents the yearly profits for a company, where P(t) is the profit in year t . (a) Is the value of a positive, negative, or zero? Explain. (b) Write an expression in terms of a and b that represents the year t when the company made the least profit. (c) The company made the same yearly profits in 2004 and 2012 . Estimate the year in which the company made the least profit. (d) Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or constant? Explain. Check back soon! ### Problem 95 Proof Assume that the function$$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 $f$ .