Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 11

Integration - all with Video Answers

Educators

Section 1

Antiderivatives and the Rules of Integration

Problem 1

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=\frac{1}{3} x^{3}+2 x^{2}-x+2 ; f(x)=x^{2}+4 x-1$$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=x e^{x}+\pi ; f(x)=e^{x}(1+x)$$

Lucas Finney

Numerade Educator

Problem 3

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=\sqrt{2 x^{2}-1} ; f(x)=\frac{2 x}{\sqrt{2 x^{2}-1}}$$

Lucas Finney

Numerade Educator

Problem 4

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=x \ln x-x ; f(x)=\ln x$$

Lucas Finney

Numerade Educator

Problem 5

(a) verify that $G$ is an antiderivative of $f$, (b) find all antiderivatives of $f$, and $(c)$ sketch the graphs of a few of the family of antiderivatives found in part (b).
$$G(x)=2 x, f(x)=2$$

Gregory Higby

Numerade Educator

Problem 6

(a) verify that $G$ is an antiderivative of $f$, (b) find all antiderivatives of $f$, and $(c)$ sketch the graphs of a few of the family of antiderivatives found in part (b).
$$G(x)=2 x^{2} ; f(x)=4 x$$

Gregory Higby

Numerade Educator

Problem 7

(a) verify that $G$ is an antiderivative of $f$, (b) find all antiderivatives of $f$, and $(c)$ sketch the graphs of a few of the family of antiderivatives found in part (b).
$$G(x)=\frac{1}{3} x^{3} ; f(x)=x^{2}$$

Gregory Higby

Numerade Educator

Problem 8

(a) verify that $G$ is an antiderivative of $f$, (b) find all antiderivatives of $f$, and $(c)$ sketch the graphs of a few of the family of antiderivatives found in part (b).
$$G(x)=e^{x} ; f(x)=e^{x}$$

Gregory Higby

Numerade Educator

Problem 9

Find the indefinite integral.
$$\int 6 d x$$

Nishant Tyagi

Numerade Educator

Problem 10

Find the indefinite integral.
$$\int \sqrt{2} d x$$

Nishant Tyagi

Numerade Educator

Problem 11

Find the indefinite integral.
$$\int x^{3} d x$$

Nishant Tyagi

Numerade Educator

Problem 12

Find the indefinite integral.
$$\int 2 x^{5} d x$$

Nishant Tyagi

Numerade Educator

Problem 13

Find the indefinite integral.
$$\int x^{-4} d x$$

Nishant Tyagi

Numerade Educator

Problem 14

Find the indefinite integral.
$$\int 3 t^{-7} d t$$

Nishant Tyagi

Numerade Educator

Problem 15

Find the indefinite integral.
$$\int x^{2 / 3} d x$$

Gregory Higby

Numerade Educator

Problem 16

Find the indefinite integral.
$$\int 2 u^{3 / 4} d u$$

Nishant Tyagi

Numerade Educator

Problem 17

Find the indefinite integral.
$$\int x^{-5 / 4} d x$$

Nishant Tyagi

Numerade Educator

Problem 18

Find the indefinite integral.
$$\int 3 x^{-2 / 3} d x$$

Nishant Tyagi

Numerade Educator

Problem 19

Find the indefinite integral.
$$\int \frac{2}{x^{2}} d x$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 20

Find the indefinite integral.
$$\int \frac{1}{3 x^{5}} d x$$

Nishant Tyagi

Numerade Educator

Problem 21

Find the indefinite integral.
$$\int \pi \sqrt{t} d t$$

Nishant Tyagi

Numerade Educator

Problem 22

Find the indefinite integral.
$$\int \frac{3}{\sqrt{t}} d t$$

Nishant Tyagi

Numerade Educator

Problem 23

Find the indefinite integral.
$$\int(3-2 x) d x$$

Linda Hand

Numerade Educator

Problem 24

Find the indefinite integral.
$$\int\left(1+u+u^{2}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 25

Find the indefinite integral.
$$\int\left(x^{2}+x+x^{-3}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 26

Find the indefinite integral.
$$\int\left(0.3 t^{2}+0.02 t+2\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 27

Find the indefinite integral.
$$\int 4 e^{x} d x$$

Sirat Shah

Numerade Educator

Problem 28

Find the indefinite integral.
$\int\left(1+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 29

Find the indefinite integral.
$$\int\left(1+x+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 30

Find the indefinite integral.
$$\int\left(2+x+2 x^{2}+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 31

Find the indefinite integral.
$$\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 32

Find the indefinite integral.
$$\int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 33

Find the indefinite integral.
$$\int\left(x^{5 / 2}+2 x^{3 / 2}-x\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 34

Find the indefinite integral.
$$\int\left(t^{3 / 2}+2 t^{1 / 2}-4 t^{-1 / 2}\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 35

Find the indefinite integral.
$$\int\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right) d x$$

Gregory Higby

Numerade Educator

Problem 36

Find the indefinite integral.
$$\int\left(\sqrt[3]{x^{2}}-\frac{1}{x^{2}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 37

Find the indefinite integral.
$$\int\left(\frac{u^{3}+2 u^{2}-u}{3 u}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 38

Find the indefinite integral.
$$\int \frac{x^{4}-1}{x^{2}} d x$$

Nishant Tyagi

Numerade Educator

Problem 39

Find the indefinite integral.
$$\int(2 t+1)(t-2) d t$$

Nishant Tyagi

Numerade Educator

Problem 40

Find the indefinite integral.
$$\int u^{-2}\left(1-u^{2}+u^{4}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 41

Find the indefinite integral.
$$\int \frac{1}{x^{2}}\left(x^{4}-2 x^{2}+1\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 42

Find the indefinite integral.
$$\int \sqrt{t}\left(t^{2}+t-1\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 43

Find the indefinite integral.
$$\int \frac{d s}{(s+1)^{-2}}$$

Nishant Tyagi

Numerade Educator

Problem 44

Find the indefinite integral.
\$$\int\left(\sqrt{x}+\frac{3}{x}-2 e^{x}\right) d x$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 45

Find the indefinite integral.
$$\int\left(e^{t}+t^{e}\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 46

Find the indefinite integral.
$$\int\left(\frac{1}{x^{2}}-\frac{1}{\sqrt[3]{x^{2}}}+\frac{1}{\sqrt{x}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 47

Find the indefinite integral.
$$\int\left(\frac{x^{3}+x^{2}-x+1}{x^{2}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 48

Find the indefinite integral.
$$\int \frac{t^{3}+\sqrt[3]{t}}{t^{2}} d t$$

Nishant Tyagi

Numerade Educator

Problem 49

Find the indefinite integral.
$$\int \frac{(\sqrt{x}-1)^{2}}{x^{2}} d x$$

Nishant Tyagi

Numerade Educator

Problem 50

Find the indefinite integral.
$$\int(x+1)^{2}\left(1-\frac{1}{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 51

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=2 x+1 ; f(1)=3$$

Nishant Tyagi

Numerade Educator

Problem 52

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=3 x^{2}-6 x ; f(2)=4$$

Nishant Tyagi

Numerade Educator

Problem 53

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=3 x^{2}+4 x-1 ; f(2)=9$$

Nishant Tyagi

Numerade Educator

Problem 54

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=\frac{1}{\sqrt{x}} ; f(4)=2$$

Nishant Tyagi

Numerade Educator

Problem 55

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=1+\frac{1}{x^{2}} ; f(1)=2$$

Nishant Tyagi

Numerade Educator

Problem 56

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=e^{x}-2 x ; f(0)=2$$

Nishant Tyagi

Numerade Educator

Problem 57

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=\frac{x+1}{x} ; f(1)=1$$

Nishant Tyagi

Numerade Educator

Problem 58

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=1+e^{x}+\frac{1}{x} ; f(1)=3+e$$

Nishant Tyagi

Numerade Educator

Problem 59

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{1}{2} x^{-1 / 2} ;(2, \sqrt{2})$$
60. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Gregory Higby

Numerade Educator

Problem 60

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Gregory Higby

Numerade Educator

Problem 61

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=e^{x}+x ;(0,3)$$

Gregory Higby

Numerade Educator

Problem 62

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{2}{x}+1 ;(1,2)$$

Gregory Higby

Numerade Educator

Problem 63

Madison Finance opened two branches on September $1(t=0)$. Branch $\mathrm{A}$ is located in an established industrial park, and branch B is located in a fast-growing new development. The net rate at which money was deposited into branch $A$ and branch $B$ in the first 180 business days is given by the graphs of $f$ and $g$, respectively (see the figure). Which branch has a larger amount on deposit at the end of 180 business days? Justify your answer.

Lucas Finney

Numerade Educator

Problem 64

Two cars, side by side, start from rest and travel along a straight road. The velocity of car $\mathrm{A}$ is given by $v=f(t)$, and the velocity of car $\mathrm{B}$ is given by $v=$ $g(t)$. The graphs of $f$ and $g$ are shown in the figure below. Are the cars still side by side after $T \mathrm{sec} ?$ If not, which car is ahead of the other? Justify your answer.

Lucas Finney

Numerade Educator

Problem 65

The velocity of a car (in feet/second) $t$ sec after starting from rest is given by the function
$$
f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30)
$$
Find the car's position, $s(t)$, at any time $t$. Assume $s(0)=0$.

Lucas Finney

Numerade Educator

Problem 66

The velocity of a car (in feet/second) $t$ sec after starting from rest is given by the function
$$
f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30)
$$
Find the car's position, $s(t)$, at any time $t$. Assume $s(0)=0$.

Lucas Finney

Numerade Educator

Problem 66

The velocity (in feet/second) of a maglev is
$$
v(t)=0.2 t+3 \quad(0 \leq t \leq 120)
$$
At $t=0$, it is at the station. Find the function giving the position of the maglev at time $t$, assuming that the motion takes place along a straight stretch of track.

Gregory Higby

Numerade Educator

Problem 67

Lorimar Watch Company manufactures travel clocks. The daily marginal cost function associated with producing these clocks is
$$
C^{\prime}(x)=0.000009 x^{2}-0.009 x+8
$$
where $C^{\prime}(x)$ is measured in dollars/unit and $x$ denotes the number of units produced. Management has determined that the daily fixed cost incurred in producing these clocks is $\$ 120 .$ Find the total cost incurred by Lorimar in producing the first 500 travel clocks/day.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 68

The management of Lorimar Watch Company has determined that the daily marginal revenue function associated with producing and selling their travel clocks is given by
$$
R^{\prime}(x)=-0.009 x+12
$$
where $x$ denotes the number of units produced and sold and $R^{\prime}(x)$ is measured in dollars/unit.
a. Determine the revenue function $R(x)$ associated with producing and selling these clocks.
b. What is the demand equation that relates the wholesale unit price with the quantity of travel clocks demanded?

Lucas Finney

Numerade Educator

Problem 69

Cannon Precision Instruments makes an automatic electronic flash with Thyrister circuitry. The estimated marginal profit associated with producing and selling these electronic flashes is
$$
P^{\prime}(x)=-0.004 x+20
$$
dollars/unit/month when the production level is $x$ units per month. Cannon's fixed cost for producing and selling these electronic flashes is $\$ 16,000 /$ month. At what level of production does Cannon realize a maximum profit? What is the maximum monthly profit?

Lucas Finney

Numerade Educator

Problem 70

Carlota Music Company estimates that the marginal cost of manufacturing its Professional Series guitars is
$$
C^{\prime}(x)=0.002 x+100
$$
dollars/month when the level of production is $x$ guitars/ month. The fixed costs incurred by Carlota are $\$ 4000$ / month. Find the total monthly cost incurred by Carlota in manufacturing $x$ guitars/month.

Chris Trentman

Numerade Educator

Problem 71

The national health expenditures are projected to grow at the rate of
$$
r(t)=0.0058 t+0.159 \quad(0 \leq t \leq 13)
$$
trillion dollars/year from 2002 through $2015 .$ Here, $t=0$ corresponds to 2002. The expenditure in 2002 was $$\$ 1.60$$ trillion.
a. Find a function $f$ giving the projected national health expenditures in year $t$.
b. What does your model project the national health expenditure to be in 2015 ?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 72

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector $t$ hr into the $8 \mathrm{a} . \mathrm{m}$. to 12 noon morning shift is approximately
$$
N^{\prime}(t)=-3 t^{2}+12 t+45 \quad(0 \leq t \leq 4)
$$
a. Find an expression $N(t)$ that approximates the number of sets inspected at the end of $t$ hours. Hint: $N(0)=0$.
b. How many sets does the average inspector check during a morning shift?

Sheryl Ezze

Numerade Educator

Problem 73

Based on data obtained by polling automobile buyers, the number of subscribers of satellite radios is expected to grow at the rate of
$$
r(t)=-0.375 t^{2}+2.1 t+2.45 \quad(0 \leq t \leq 5)
$$
million subscribers/year between $2003(t=0)$ and 2008 $(t=5)$. The number of satellite radio subscribers at the beginning of 2003 was $1.5$ million.
a. Find an expression giving the number of satellite radio subscribers in year $t(0 \leq t \leq 5)$.
b. Based on this model, what was the number of satellite radio subscribers in 2008 ?

Carson Merrill

Numerade Educator

Problem 74

The rate at which the risk of Down syndrome is changing is approximated by the function
$r(x)=0.004641 x^{2}-0.3012 x+4.9 \quad(20 \leq x \leq 45)$
where $r(x)$ is measured in percentage of all births/year and $x$ is the maternal age at delivery.
a. Find a function $f$ giving the risk as a percentage of all births when the maternal age at delivery is $x$ years, given that the risk of down syndrome at age 30 is $0.14 \%$ of all births.
b. Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 40 years? 45 years?

Lucas Finney

Numerade Educator

Problem 75

The average credit card debt per U.S. household between $1990(t=0)$ and $2003(t=13)$ was growing at the rate of approximately
$D(t)=-4.479 t^{2}+69.8 t+279.5 \quad(0 \leq t \leq 13)$
dollars/year. The average credit card debt per U.S. household stood at $$\$ 2917$$ in 1990 .
a. Find an expression giving the approximate average credit card debt per U.S. household in year $t$ $(0 \leq t \leq 13) .$
b. Use the result of part (a) to estimate the average credit card debt per U.S. household in 2003 .

Anurag Kumar

Numerade Educator

Problem 76

The total number of acres of genetically modified crops grown worldwide from 1997 through 2003 was changing at the rate of
$$
R(t)=2.718 t^{2}-19.86 t+50.18 \quad(0 \leq t \leq 6)
$$
million acres/year. The total number of acres of such crops grown in 1997 ( $t=0$ ) was $27.2$ million acres. How many acres of genetically modified crops were grown worldwide in $2003 ?$

Lucas Finney

Numerade Educator

Problem 77

One method of weight loss gaining in popularity is stomach-reducing surgery. It is generally reserved for people at least $100 \mathrm{lb}$ overweight because the procedure carries a serious risk of death or complications. According to the American Society of Bariatric Surgery, the number of morbidly obese patients undergoing the procedure was increasing at the rate of
$$
R(t)=9.399 t^{2}-13.4 t+14.07 \quad(0 \leq t \leq 3)
$$
thousands/year, with $t=0$ corresponding to 2000 . The number of gastric bypass surgeries performed in 2000 was $36.7$ thousand.
a, Find an expression giving the number of gastric bypass surgeries performed in year $t(0 \leq t \leq 3)$.
b. Use the result of part (a) to find the number of gastric bypass surgeries performed in 2003 .

Hossam Mohamed

Numerade Educator

Problem 78

According to a study conducted in 2004, the share of online advertisement, worldwide, as a percentage of the total ad market, is expected to grow at the rate of
$$
R(t)=-0.033 t^{2}+0.3428 t+0.07 \quad(0 \leq t \leq 6)
$$
percent/year at time $t$ (in years), with $t=0$ corresponding to the beginning of 2000 . The online ad market at the beginning of 2000 was $2.9 \%$ of the total ad market.
a. What is the projected online ad market share at any time $t$ ?
b. What was the projected online ad market share at the beginning of 2005 ?

Willis James

Numerade Educator

Problem 79

The average out-of-pocket costs for beneficiaries in traditional Medicare (including premiums, cost sharing, and prescription drugs not covered by Medicare) is projected to grow at the rate of
$$
C^{\prime}(t)=12.288 t^{2}-150.5594 t+695.23
$$
dollars/year, where $t$ is measured in 5 -yr intervals, with $t=0$ corresponding to $2000 .$ The out-of-pocket costs for beneficiaries in 2000 were $$\$ 3142$$.
a. Find an expression giving the average out-of-pocket costs for beneficiaries in year $t$.
b. What is the projected average out-of-pocket costs for beneficiaries in 2010 ?

Wendi Zhao

Numerade Educator

Problem 80

A ballast is dropped from a stationary hot-air balloon that is hovering at an altitude of $400 \mathrm{ft}$. Its velocity after $t$ sec is $-32 t \mathrm{ft} / \mathrm{sec}$.
a. Find the height $h(t)$ of the ballast from the ground at time $t$ Hint: $h^{\prime}(t)=-32 t$ and $h(0)=400$.
b. When will the ballast strike the ground?
c. Find the velocity of the ballast when it hits the ground.

Lucas Finney

Numerade Educator

Problem 81

A study conducted by TeleCable estimates that the number of cable TV subscribers will grow at the rate of
$$
100+210 t^{3 / 4}
$$
new subscribers/month, $t$ mo from the start date of the service. If 5000 subscribers signed up for the service before the starting date, how many subscribers will there be $16 \mathrm{mo}$ from that date?

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 82

The rate of change of the level of ozone, an invisible gas that is an irritant and impairs breathing, present in the atmosphere on a certain May day in the city of Riverside is given by
$$
R(t)=3.2922 t^{2}-0.366 t^{3} \quad(0<t<11)
$$
(measured in pollutant standard index/hour). Here, $t$ is measured in hours, with $t=0$ corresponding to $7 \mathrm{a} . \mathrm{m}$. Find the ozone level $A(t)$ at any time $t$, assuming that at $7 \mathrm{a} \cdot \mathrm{m}$. it is zero.

Lucas Finney

Numerade Educator

Problem 83

The velocity, in feet/second, of a rocket $t$ sec into vertical flight is given by
$$
v(t)=-3 t^{2}+192 t+120
$$
Find an expression $h(t)$ that gives the rocket's altitude, in feet, $t$ sec after liftoff. What is the altitude of the rocket 30 sec after liftoff?

Lucas Finney

Numerade Educator

Problem 84

The development of AstroWorld ("The Amusement Park of the Future") on the outskirts of a city will increase the city's population at the rate of
$$
4500 \sqrt{t}+1000
$$
people/year, $t$ yr from the start of construction. The population before construction is $30,000 .$ Determine the projected population 9 yr after construction of the park has begun.

Lucas Finney

Numerade Educator

Problem 85

The sales of organic milk from 1999 through 2004 grew at the rate of approximately
$$
\begin{array}{r}
R(t)=3 t^{3}-17.9445 t^{2}+28.7222 t+26.632 \\
(0 \leq t \leq 5)
\end{array}
$$
million dollars/year, where $t$ is measured in years, with $t=0$ corresponding to $1999 .$ Sales of organic milk in 1999 totaled $$\$ 108$$ million.
a. Find an expression giving the total sales of organic milk by year $t(0 \leq t \leq 5)$.
b. According to this model, what were the total sales of organic milk in $2004 ?$

Anna Jones

Numerade Educator

Problem 86

Empirical data suggest that the surface area of a $180-\mathrm{cm}$ -tall human body changes at the rate of
$$
S^{\prime}(W)=0.131773 W^{-0.575}
$$
square meters/kilogram, where $W$ is the weight of the body in kilograms. If the surface area of a 180 -cm-tall human body weighing $70 \mathrm{~kg}$ is $1.886277 \mathrm{~m}^{2}$, what is the surface area of a human body of the same height weighing $75 \mathrm{~kg}$ ?

Breanna Ollech

Numerade Educator

Problem 87

The number of Medicarecertified home-health-care agencies ( $70 \%$ are freestanding, and $30 \%$ are owned by a hospital or other large facility) has been declining at the rate of
$$
0.186 e^{-0.02 t} \quad(0 \leq t \leq 14)
$$
thousand agencies/year between $1988(t=0)$ and 2002 $(t=14)$. The number of such agencies stood at $9.3$ thousand units in 1988 .
a. Find an expression giving the number of health-care agencies in year $t$.
b. What was the number of health-care agencies in $2002 ?$
c. If this model held true through 2005 , how many care agencies were there in 2005 ?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 88

According to the Jenss model for predicting the height of preschool children, the rate of growth of a typical preschool child is
$$
R(t)=25.8931 e^{-0.993 t}+6.39 \quad\left(\frac{1}{4} \leq t \leq 6\right)
$$
centimeters/year, where $t$ is measured in years. The height of a typical 3 -mo-old preschool child is $60.2952 \mathrm{~cm}$.
a. Find a model for predicting the height of a typical preschool child at age $t$.
b. Use the result of part (a) to estimate the height of a typical 1-yr-old child.

Lucas Finney

Numerade Educator

Problem 89

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r \mathrm{~cm}$ from the central axis of an artery (in centimeters/second/centimeter) is given by
$$
a(r)=-k r
$$
where $k$ is a constant. If the radius of an artery is $R \mathrm{~cm}$, find an expression for the velocity of blood as a function of $y$ (see the accompanying figure).

Lucas Finney

Numerade Educator

Problem 90

A car traveling along a straight road at $66 \mathrm{ft} / \mathrm{sec}$ accelerated to a speed of $88 \mathrm{ft} / \mathrm{sec}$ over $\mathrm{a}$ distance of $440 \mathrm{ft}$. What was the acceleration of the car,
assuming it was constant?

Lucas Finney

Numerade Educator

Problem 91

What constant deceleration would a car moving along a straight road have to be subjected to if it were brought to rest from a speed of $88 \mathrm{ft} / \mathrm{sec}$ in $9 \mathrm{sec}$ ? What would be the stopping distance?

Lucas Finney

Numerade Educator

Problem 92

A pilot lands a fighter aircraft on an aircraft carrier. At the moment of touchdown, the speed of the aircraft is $160 \mathrm{mph}$. If the aircraft is brought to a complete stop in 1 sec and the deceleration is assumed to be constant, find the number of $g$ 's the pilot is subjected to during landing $\left(1 \mathrm{~g}=32 \mathrm{ft} / \mathrm{sec}^{2}\right) .$

Lucas Finney

Numerade Educator

Problem 93

After rounding the final turn in the bell lap, two runners emerged ahead of the pack. When runner A is $200 \mathrm{ft}$ from the finish line, his speed is $22 \mathrm{ft} / \mathrm{sec}$, a speed that he maintains until he crosses the line. At that instant of time, runner $\mathrm{B}$, who is $20 \mathrm{ft}$ behind runner A and running at a speed of $20 \mathrm{ft} / \mathrm{sec}$, begins to sprint. Assuming that runner B sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of runner A?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 94

A tank has a constant cross-sectional area of $50 \mathrm{ft}^{2}$ and an orifice of constant cross-sectional area of $\frac{1}{2} \mathrm{ft}^{2}$ located at the bottom of the tank (see the accompanying figure).
If the tank is filled with water to a height of $h \mathrm{ft}$ and allowed to drain, then the height of the water decreases at a rate that is described by the equation
$$
\frac{d h}{d t}=-\frac{1}{25}\left(\sqrt{20}-\frac{t}{50}\right) \quad(0 \leq t \leq 50 \sqrt{20})
$$
Find an expression for the height of the water at any time $t$ if its height initially is $20 \mathrm{ft}$

Lucas Finney

Numerade Educator

Problem 95

During a thunderstorm, rain was falling at the rate of
$$
\frac{8}{(t+4)^{2}} \quad(0 \leq t \leq 2)
$$
inches/hour.
a. Find an expression giving the total amount of rainfall after $t$ hr. Hint: The total amount of rainfall at $t=0$ is zero.
b. How much rain had fallen after $1 \mathrm{hr}$ ? After $2 \mathrm{hr}$ ?

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 96

A fighter aircraft is launched from the deck of a Nimitz-class aircraft carrier with the help of a steam catapult. If the aircraft is to attain a takeoff speed of at least $240 \mathrm{ft} / \mathrm{sec}$ after traveling $800 \mathrm{ft}$ along the flight deck, find the minimum acceleration it must be subjected to, assuming it is constant.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 97

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $F$ and $G$ are antiderivatives of $f$ on an interval $I$, then $F(x)=G(x)+C$ on $I$.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 98

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $F$ is an antiderivative of $f$ on an interval $I$, then $\int f(x) d x=F(x) .$

Lucas Finney

Numerade Educator

Problem 99

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are integrable, then $\int[2 f(x)-3 g(x)] d x=$ $2 \int f(x) d x-3 \int g(x) d x$

Gregory Higby

Numerade Educator

Problem 100

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are integrable, then $\int f(x) g(x) d x=$ $\left[\int f(x) d x\right]\left[\int g(x) d x\right]$

Gregory Higby

Numerade Educator