Calculus, Early Transcendentals

Dennis G. Zill, Warren S. Wright

Chapter 6

Applications of the Integral - all with Video Answers

Educators

Section 1

Rectilinear Motion Revisited

Problem 1

In Problems $1-6,$ a body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=6 ; s=5 \text { when } t=2
$$

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

A body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=2 t+1 ; s=0 \text { when } t=1
$$

Gregory Higby

Numerade Educator

Problem 3

A body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=t^{2}-4 t ; s=6 \text { when } t=3
$$

Gregory Higby

Numerade Educator

Problem 4

A body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=\sqrt{4 t+5} ; s=2 \text { when } t=1
$$

Gregory Higby

Numerade Educator

Problem 5

A body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=-10 \cos (4 t+\pi / 6) ; s=\frac{5}{4} \text { when } t=0
$$

Gregory Higby

Numerade Educator

Problem 6

A body moves in a straight line with velocity $v(t) .$ Find the position function $s(t)$.
$$
v(t)=2 \sin 3 t ; s=0 \text { when } t=\pi
$$

Gregory Higby

Numerade Educator

Problem 7

In Problems $7-12$, a body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$
$$
a(t)=-5 ; v=4 \text { and } s=2 \text { when } t=1
$$

Gregory Higby

Numerade Educator

Problem 8

A body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$.
$$
a(t)=6 t ; v=0 \text { and } s=-5 \text { when } t=2
$$

Gregory Higby

Numerade Educator

Problem 9

A body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$.
$$
a(t)=3 t^{2}-4 t+5 ; v=-3 \text { and } s=10 \text { when } t=0
$$

Gregory Higby

Numerade Educator

Problem 10

A body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$.
$$
a(t)=(t-1)^{2} ; v=4 \text { and } s=6 \text { when } t=1
$$

Gregory Higby

Numerade Educator

Problem 11

A body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$.
$$
a(t)=7 t^{1 / 3}-1 ; v=50 \text { and } s=0 \text { when } t=8
$$

Gregory Higby

Numerade Educator

Problem 12

A body moves in a straight line with acceleration $a(t) .$ Find $v(t)$ and $s(t)$.
$$
a(t)=100 \cos 5 t ; v=-20 \text { and } s=15 \text { when } t=\pi / 2
$$

Gregory Higby

Numerade Educator

Problem 13

In Problems $13-18,$ an object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=t^{2}-2 t ; \quad[0,5]
$$

Gregory Higby

Numerade Educator

Problem 14

An object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=-t^{2}+4 t+7 ; \quad[0,6]
$$

Gregory Higby

Numerade Educator

Problem 15

An object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=t^{3}-3 t^{2}-9 t ; \quad[0,4]
$$

Gregory Higby

Numerade Educator

Problem 16

An object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=t^{4}-32 t^{2} ; \quad[1,5]
$$

Gregory Higby

Numerade Educator

Problem 17

An object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=6 \sin \pi t ; \quad[1,3]
$$

Gregory Higby

Numerade Educator

Problem 18

An object moves in a straight line according to the given position function. If $\mathrm{s}$ is measured in centimeters, find the total distance traveled by the object in the indicated time interval.
$$
s(t)=(t-3)^{2} ; \quad[2,7]
$$

Gregory Higby

Numerade Educator

Problem 19

A driver of a car that is traveling in a straight line at a constant $60 \mathrm{mi} / \mathrm{h}$ takes his eyes off the road for $2 \mathrm{~s}$. How many feet does the car move in this time?

Gregory Higby

Numerade Educator

Problem 20

A ball is dropped (released from rest) from a height of $144 \mathrm{ft}$. How long does it take for the ball to hit the ground? At what speed does it hit the ground?

Gregory Higby

Numerade Educator

Problem 21

An egg is dropped from the top of a building and hits the ground $4 \mathrm{~s}$ from release. How tall is the building?

Gregory Higby

Numerade Educator

Problem 22

A stone is dropped into a well and the splash is heard $2 \mathrm{~s}$ later. If the speed of sound in air is $1080 \mathrm{ft} / \mathrm{s},$ find the depth of the well.

Patrick Connors

Patrick Connors

Numerade Educator

Problem 23

An arrow is projected vertically upward from ground level with an initial velocity of $24.5 \mathrm{~m} / \mathrm{s}$. How high does it rise?

Gregory Higby

Numerade Educator

Problem 24

How high would the arrow in Problem 23 rise on the planet Mars where $g=3.6 \mathrm{~m} / \mathrm{s}^{2} ?$

Gregory Higby

Numerade Educator

Problem 25

A golf ball is thrown vertically upward from the edge of the roof of a 384 -ft-high building with an initial velocity of $32 \mathrm{ft} / \mathrm{s}$. What is the maximum height attained by the ball? At what time does the ball hit the ground?

Gregory Higby

Numerade Educator

Problem 26

In Problem $25,$ what is the velocity of the golf ball as it passes an observer in a window that is $256 \mathrm{ft}$ off the ground?

Gregory Higby

Numerade Educator

Problem 27

A person throws a marshmallow vertically downward with an initial velocity of $16 \mathrm{ft} / \mathrm{s}$ from a window that is $102 \mathrm{ft}$ off the ground. If the marshmallow hits a 6 -ft-tall person on the head, what is the impact velocity?

Gregory Higby

Numerade Educator

Problem 28

The person hit on the head in Problem 27 climbs to the top of a $22-\mathrm{ft}$ -high ladder and throws a stone vertically upward with an initial velocity of $96 \mathrm{ft} / \mathrm{s}$. If the stone hits the culprit at the 102 -ft level, what is the impact velocity?

Gregory Higby

Numerade Educator

Problem 29

In March $1979,$ the Voyager 1 space probe photographed an active volcanic eruption on Io, one of the moons of Jupiter. Find the ejection velocity of a rock from the volcano Loki if the rock attains a height of $200 \mathrm{~km}$ above the summit of the volcano. On Io the acceleration due to gravity is $g=1.8 \mathrm{~m} / \mathrm{s}^{2}$

Ryan Mcclanahan

Ryan Mcclanahan

Numerade Educator

Problem 30

As shown at a point $30 \mathrm{ft}$ from a $25-\mathrm{ft}$ -tall street lamp a ball is thrown vertically downward from a height of $25 \mathrm{ft}$ with an initial velocity of $2 \mathrm{ft} / \mathrm{s}$.
(a) Find the rate at which the shadow of the ball is moving toward the base of the street lamp.
(b) Find the rate at which the shadow of the ball is moving toward the base of the street lamp at $t=\frac{1}{2}$.

Fasiha Binat Zafar

Fasiha Binat Zafar

Numerade Educator

Problem 31

If a body is moving rectilinearly with a constant acceleration $a$ and $v=v_{0}$ when $s=0$, show that
$$
v^{2}=v_{0}^{2}+2 a s .\left[\text { Hint: } \frac{d v}{d t}=\frac{d v}{d s} \frac{d s}{d t}=\frac{d v}{d s} v .\right]
$$

Doruk Isik

Numerade Educator

Problem 32

Show that, when air resistance is ignored, a projectile shot vertically upward from ground level hits the ground again with a speed equal to the initial velocity $v_{0}$.

Gregory Higby

Numerade Educator

Problem 33

Suppose the acceleration due to gravity on a planet is onehalf that on the Earth. Prove that a ball tossed vertically upward from the surface of the planet would attain a maximum height twice that on the Earth when the same initial velocity is used.

Donald Albin

Numerade Educator

Problem 34

In Problem $33,$ suppose the initial velocity of the ball on the planet is $v_{0}$ and the initial velocity of the ball on the Earth is $2 v_{0}$. Compare the maximum heights attained. Determine the initial velocity of the ball on the Earth (in terms of $v_{0}$ ) so that the maximum height attained is the same as on the planet.

Carson Merrill

Numerade Educator