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Problem 72

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Problem 68

A sporting goods manufacturer designs a golf ball having a : volume of 2.48 cubic inches.

(a) What is the radius of the golf ball?

(b) The volume of the golf ball varies between 2.45 cubic inches. How does the radius vary?

(c) Use the $\varepsilon-\delta$ definition of limit to describe this situation. Identify $\varepsilon$ and $\delta .$

Answer

$\epsilon=\frac{0.843-0.836}{2}=0.0035$

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## Recommended Questions

Three tennis balls are stored in a cylindrical container with a height of 8 inches and a radius of 1.43 inches. The circumference of a tennis ball is 8 inches.

a. Find the volume of a tennis ball.

b. Find the amount of space within the cylinder not taken up by the tennis balls.

The volume of a sphere is found with the formula $V=\frac{4}{3} \pi r^{3},$ where $r$ is the length of the radius of the sphere.

A ball in the shape of a sphere has a volume of $288 \pi$ in.3. What is the radius of the ball?

(FIGURE CANNOT COPY)

A spherical steel ball bearing has a diameter of 2.540 $\mathrm{cm}$ at $25.00^{\circ} \mathrm{C}$ (a) What is its diameter when its temperature is raised to $100.0^{\circ} \mathrm{C}$ ? (b) What temperature change is required to increase its volume by 1.000$\%$ ?

A spherical steel ball bearing has a diameter of 2.540 $\mathrm{cm}$ at $25.00^{\circ} \mathrm{C}$ . (a) What is its diameter when its temperature is raised to $100.0^{\circ} \mathrm{C}$ ? (b) What temperature change is required to increase its volume by 1.000$\%$ ?

The diameter of a sphere is measured to be 5.36 in. Find (a) the radius of the sphere in centimeters, (b) the surface area of the sphere in square centimeters, and (c) the volume of the sphere in cubic centimeters.

A solid metal ball with radius 8 $\mathrm{cm}$ is melted down and recast as a solid cone with the same radius.

a. What is the height of the cone?

b. Use a calculator to show that the lateral area of the cone is about 356 more than the area of the sphere.

The shot (a metal sphere) used in the women's shot put has a volume of about 524 cubic centimeters. Find the radius of the shot.

The Hoberman Sphere is a toy ball that expands and contracts. When it is completely closed, it has a diameter of 9.5 inches. Find the volume of the Hoberman Sphere when it is completely closed. Use 3.14 for $\pi .$ Round to the nearest whole cubic inch. (Source: Hoberman Designs, Inc.)

STANDARDIZED TEST PRACTICE An NCAA (National Collegiate Athletic Association) basketball has a radius of $4 \frac{3}{4}$ inches. What is its surface area?

$A \frac{361 \pi}{16} \mathrm{in}^{2}$

$B \frac{361 \pi}{4} \mathrm{in}^{2}$

C $19 \pi$ in $^{2}$

D $361 \pi$ in $^{2}$

A hollow rubber ball has outer radius 11 $\mathrm{cm}$ and inner radius 10 $\mathrm{cm} .$

a. Find the exact volume of the rubber. Then evaluate the volume to the nearest cubic centimeter.

b. The volume of the rubber can be approximated by the formula: Use this formula to approximate $V$ . Compare your answer with the answer in part (a).

C. Is the approximation method used in part (b) better for a ball with a thick layer of rubber or a ball with a thin layer?

A sphere is inscribed in a cube with a volume of 64 cubic inches. What is the surface area of the sphere? Explain your reasoning.

The volume $V$ (in cubic inches) of a can that is 4 in. tall is given by the equation $V=4 \pi r^{2},$ where $r$ is the radius

of the can, measured in inches.

a. Solve the equation for $r .$ Do not rationalize the denominator.

b. Using the equation from part (a), determine the radius of a can with a volume of 12.56 in.' Use 3.14 for $\pi .$

Multiple Choice The circumference of a basketball for college women must be from 28.5 in. to 29.0 in. Which absolute value inequality best represents the circumference of the ball?

(A) $|C-0.25| \geq 28.5$

(B) $|C-0.25| \leq 29.0$

(C) $|C-28.75| \leq 0.25$

(D) $|C-28.75| \geq 0.25$

Ball's changing volume ume of a ball $\left(V=(4 / 3) \pi r^{3}\right)$ with respect to the radius when the

radius is $r=2 ?$

Ball's changing volume

ume of a ball $\left(V=(4 / 3) \pi r^{3}\right)$ with respect to the radius when

the radius is $r=2 ?$

A bouncing ball reaches heights of $16 \mathrm{cm}, 12.8 \mathrm{cm},$ and 10.24 $\mathrm{cm}$ on three consecutive bounces.

a. If the ball started at a height of $25 \mathrm{cm},$ how many times has it bounced when it reaches a height of 16 $\mathrm{cm} ?$

b. Write a geometric series for the downward distances the ball travels from its release at 25 $\mathrm{cm} .$

c. Write a geometric series for the upward distances the ball travels from its first bounce.

d. Find the total vertical distance the ball travels before it comes to rest.

The radius $r$ of a sphere that has volume $V$ is $r=\sqrt{\frac{3 V}{4}}$. The volume of a basketball is approximately 434.67 in 3 . The radius of a tennis ball is about one fourth the radius of a basketball. Find the radius of the tennis ball.

For Exercises 33 and $34,$ use the following information. A women's regulation-sized basketball is slightly smaller than a men's basketball. The radius $r$ of the ball that holds $V$ volume of air is $r=\left(\frac{3 V}{4 \pi}\right)^{\frac{1}{3}}$.

Find the radius of a men's basketball if it will hold 455 cubic inches of air.

The volume of a mini-basketball is about 230 cubic inches. What is its radius? Round to the nearest inch.

This problem will prepare you for the Concept Connection on page 724 . A company sells orange juice in spherical containers that look like oranges. Each container has a surface area of approximately 50.3 in $^{2}$.

a. What is the volume of the container? Round to the nearest tenth.

b. The company decides to increase the radius of the container by $10 \%$. What is the volume of the new container?

When the Hoberman Sphere (see Exercise 65 ) is completely expanded, its diameter is 30 inches. Find the volume of the Hoberman Sphere when it is completely expanded. Use 3.14 for $\pi .$ Round to the nearest whole cubic inch. (Source: Hoberman Designs, Inc.)

In the following exercises, find the

a volume b surface area of the sphere.

a baseball with radius 1.45 inches

For Exercises 33 and $34,$ use the following information. A women's regulation-sized basketball is slightly smaller than a men's basketball. The radius $r$ of the ball that holds $V$ volume of air is $r=\left(\frac{3 V}{4 \pi}\right)^{\frac{1}{3}}$.

Find the radius of a women's basketball if it will hold 413 cubic inches of air.

The surface area of a sphere is given by $A(r)=4 \pi r^{2},$ where $r$ is in inches and $A(r)$ is in square inches. The function $C(x)=6.4516 x$ takes $x$ square inches as input and outputs the equivalent result in square centimeters. Find $(C \circ A)(r)$ and explain what it represents.

The volume of a sphere is found with the formula $V=\frac{4}{3} \pi r^{3},$ where $r$ is the length of the radius of the sphere.

Suppose that the volume of the ball described in Exercise 123 is multiplied by 8. How is the radius affected?

Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of 2.5 inches. A waffle cone is 7 inches tall and has a diameter of 3.25 inches. To the nearest hundredth,

a. find the volume of the regular ice cream cone.

b. find the volume of the waffle cone.

c. how much more ice cream fits in the waffle cone compared to the regular cone?

A golfer hits a drive 260 yards on a hole that is 400 yards long. The shot is $15^{\circ}$ off target.

a What is the distance x from the golfer's ball to the $\square$hole?

b. Assume the golfer is able to hit the ball precisely the distance found in part (a). What is the maximum angle $\theta$ (theta) by which the ball can be off target in order to land no more than 10 yards from the hole?

Volume The radius $r$ of a sphere is increasing at a rate of 3 inches per minute.

(a) Find the rates of change of the volume when $r=9$ inches and $r=36$ inches.

(b) Explain why the rate of change of the volume of the sphere is not constant even though $d r / d t$ is constant.

A machinist is required to manufacture a circular metal disk with area 1000 $ cm^2 $.

(a) What radius produces such a disk?

(b) If the machinist is allowed an error tolerance of $ \pm 5 cm^2 $ in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius?

(c) In terms of the $ \varepsilon $, $ \delta $ definition of $ \displaystyle \lim_{x \to a} f(x) = L $, what is $ x $? What is $ f(x) $? What is $ a $? What is $ L $? What value of $ \varepsilon $ is given? What is the corresponding value of $ \delta $?

What is the volume, in cubic inches, of a cube whose total surface

area is 216 square inches?

\begin{equation}

\begin{array}{l}{\text { (A) } 18} \\ {\text { (B) } 36}\\{\text { (C) } 216} \\ {\text { (D) } 1,296}\end{array}

\end{equation}

A particular sphere has the property that its surface area has the same numerical value as its volume. What is the length of the radius of this sphere?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 6

The density of gold is 19.3 $\mathrm{g} / \mathrm{cm}^{3}$ .

a. What is the volume, in cubic centimeters, of a

sample of gold that has a mass of 0.715 $\mathrm{kg}$ ?

b. If this sample of gold is a cube, what is the length of each edge in centimeters?

$\bullet$$\bullet$ A player bounces a basketball on the floor, compressing it to 80.0$\%$ of its original volume. The air (assume it is essentially $\mathrm{N}_{2}$ gas) inside the ball is originally at a temperature of $20.2^{\circ} \mathrm{C}$ and a pressure of 2.00 atm. The ball's diameter is 23.9 $\mathrm{cm} .$ (a) What temperature does the air in the ball reach at

its maximum compression? (b) By how much does the internal energy of the air change between the ball's original state and its maximum compression?

Find the volume of a sphere that is circumscribed about a cube with a volume of 216 cubic inches.

Calculate the depth to which Avogadro's number of table tennis balls would cover Earth. Each ball has a diameter of $3.75 \mathrm{cm} .$ Assume the space between balls adds an extra $25.0 \%$ to their volume and assume they are not crushed by their own weight.

The density of gold is 19.3 $\mathrm{g} / \mathrm{cm}^{3} .$

a. What is the volume, in cubic centimeters,

of a sample of gold with mass 0.715 $\mathrm{kg}$ ?

b. If this sample of gold is a cube, how long

is each edge in centimeters?

Find each measurement. Give your answers in terms of $\pi$.

the radius of a sphere with volume $288 \pi \mathrm{cm}^{3}$

Volume A spherical balloon is being inflated. Find the approximate change in volume if the radius increases from 4 $\mathrm{cm}$ to 4.2 $\mathrm{cm}.$

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 $\mathrm{cm} / \mathrm{s}$ .

(a) Express the radius $r$ of the balloon as a function of the time $t($ in seconds).

(b) If $V$ is the volume of the balloon as a function of the radius, find $V \circ r$ and interpret it.

A spherical balloon is being inflated and the radius of the

balloon is increasing at a rate of 2 $\mathrm{cm} / \mathrm{s}$ .

(a) Express the radius $r$ of the balloon as a function of the

time $t($ in seconds).

(b) If $V$ is the volume of the balloon as a function of the

radius, find $V \circ r$ and interpret it.

A spherical balloon with radius $r$ inches has volume $V(r)=\frac{4}{3} \pi r^{3} .$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r+1$ inches.

Volume of a Cup Sunbelt Paper Products makes conical

paper cups by cutting a 4 -in. circular piece of paper on the radius and overlapping the paper by an angle $\alpha$ as shown in the accompanying figure.

a. Find the volume of the cup if $\alpha=30^{\circ}$

b. Write the volume of the cup as a function of $\alpha .$

(IMAGE CANNOT COPY)

Troy inflates a spherical balloon to a circumference of about 14 inches. He then adds more air to the balloon until the circumference is about 18 inches. What volume of air was added to the balloon?

a. Find the volume, correct to the nearest cubic centimeter, of a sphere inscribed in a cube with edges 6 $\mathrm{cm}$ long. Use $\pi \approx 3.14$

b. Find the volume of the region inside the cube but outside the sphere.

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s.

(a) Express the radius $ r $ of the balloon as a function of the time $ t $ (in seconds).

(b) If $ V $ is the volume of the balloon as a function of the radius, find $ V \circ r $ and interpret it.

MMH A golfer imparts a speed of 30.3 m/s to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation. (a) How much time does the ball spend in the air? (b) What is the longest hole in one that the golfer can make, if the ball does not roll when it hits the green?

Hot Air Balloons A hot air balloon is in the shape of a sphere. Approximately how much fabric was used to construct the balloon if its diameter is $32 \mathrm{ft}$ ? Round to the nearest whole number.

The length of the radius of the sphere $x^{2}+y^{2}+z^{2}+2 x-4 y=10$ is

(A) 3.16

(B) 3.38

(C) 3.46

(D) 3.74

(E) 3.87

(GRAPH CANNOT COPY)

A sphere with radius $8 \mathrm{cm}$ is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere.

(A) $2: \frac{1}{3} \pi$

(B) $2: 3 \pi$

(C) $1: \frac{4}{3} \pi$

(D) $1: \frac{2}{3} \pi$

Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.

A golf ball with an initial angle of $34^{\circ}$ lands exactly

$240 \mathrm{m}$ down the range on a level course.

a. Neglecting air friction, what initial speed would achieve this result?

b. Using the speed determined in item (a), find the maximum height reached by the ball.

Find the volume of a sphere whose radius is 9 inches. Use $\pi=3.14$.

(a) 28.26 in. $^{3}$

(b) 3052.08 in. $^{3}$

(c) 972 in. $^{3}$

(d) 2289.06 in. $^{3}$

Volume A cubical crystal is growing in size. Find the approximate change in the length of a side when the volume increases from 27 cubic mm to 27.1 cubic mm.

Diamonds are measured in carats, and 1 carat $=0.200 \mathrm{g} .$ The density of diamond is 3.51 $\mathrm{g} / \mathrm{cm}^{3}$ .

a. What is the volume of a 5.0 -carat diamond?

b. What is the mass in carats of a diamond measuring 2.8 $\mathrm{mL} ?$

(III) $(a)$ Determine a formula for the change in surface area of a uniform solid sphere of radius $r$ if its coefficient of linear expansion is $\alpha$ (assumed constant) and its tempera-ture is changed by $\Delta T .(b)$ What is the increase in area of a solid iron sphere of radius 60.0 $\mathrm{cm}$ if its temperature is raised from $15^{\circ} \mathrm{C}$ to $275^{\circ} \mathrm{C}$ ?

Conical Cup A conical cup is made from a circular piece of paper with radius $6 \mathrm{cm}$ by cutting out a sector and joining the edges as shown below. Suppose $\theta=5 \pi / 3$.

(a) Find the circumference $C$ of the opening of the cup.

(b) Find the radius $r$ of the opening of the cup. [Hint: Use $C=2 \pi r .]$

(c) Find the height $h$ of the cup. [Hint: Use the Pythagorean Theorem. $]$

(d) Find the volume of the cup.

A laboratory supply company produces graduated

cylinders, each with an internal radius of 2 inches

and an internal height between 7.75 inches and

8 inches. What is one possible volume, rounded to

the nearest cubic inch, of a graduated cylinder

produced by this company?

Manufacturing A manufacturer drills a hole through the

center of a metal sphere of radius $R .$ The hole has a radius $r$

Find the volume of the resulting ring.

Communications Satellites. Engineers have determined that a spherical communications satellite needs to have a capacity of 565.2 cubic feet to house all of its operating systems. The volume $V$ of a sphere is related to its radius $r$ by the formula $r=\sqrt[3]{\frac{3 V}{4 \pi}} .$ What radius must the satellite have to meet the engineer's specification? Use 3.14 as an approximation of $\pi$.

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0$^\circ$C. Assume that all the gas is N$_2$, of molar mass 28.0 g/mol. (a) Find the mass of a single N$_2$ molecule. (b) How much translational kinetic energy does an average N$_2$ molecule have? (c) How many N$_2$ molecules are in this balloon? (d) What is the $total$ translational kinetic energy of all the molecules in the balloon?

Measurement The volume in cubic feet of a CD holder can be expressed as $V(x)=-x^{3}-x^{2}+6 x,$ or, when factored, as the product of its three dimensions. The depth is expressed as $2-x$ . Assume that the height is greater than the width.

a. Factor the polynomial to find linear expressions for the height and the width.

b. Graph the function. Find the $x$ -intercepts. What do they represent?

c. Describe a realistic domain for the function.

d. Find the maximum volume of the CD holder.

The Classical Bead Problem A round hole is drilled

through the center of a spherical solid of radius $r .$ The resulting

cylindrical hole has height 4 $\mathrm{cm} .$

(a) What is the volume of the solid that remains?

(b) What is unusual about the answer?

A golf ball is struck at ground level. The speed of

the golf ball as a function of

the time is shown in Fig. $4-36$

where $t=0$ at the instant the ball is struck. The scaling on

the vertical axis is set by

$v_{a}=19 \mathrm{m} / \mathrm{s}$ and $v_{b}=31 \mathrm{m} / \mathrm{s}$

(a) How far does the golf

ball travel horizontally before returning to ground

level? (b) What is the maximum height above ground level attained by the ball?

A player bounces a basketball on the floor, compressing it to 80.0% of its original volume. The air (assume it is essentially N$_2$ gas) inside the ball is originally at 20.0$^\circ$C and 2.00 atm. The ball's inside diameter is 23.9 cm. (a) What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal. (b) By how much does the internal energy of the air change between the ball's original state and its maximum compression?

Earth is approximately a sphere of radius $6.37 \times 10^{6} \mathrm{m}$

What are (a) its circumference in kilometers, (b) its surface area in

square kilometers, and (c) its volume in cubic kilometers?

Volume of a Football A football is in the shape of a prolate spheroid, which is simply a solid obtained by rotating an ellipse $\left(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\right)$ about its major axis. An inflated NFL football averages 11.125 inches in length and 28.25 inches in center circumference. If the volume of a prolate spheroid is $\frac{4}{3} \pi a b^{2},$ how much air does the football contain? (Neglect material thickness).

An atom of rhodium (Rh) has a diameter of about $2.7 \times 10^{-8} \mathrm{cm} .$ (a) What is the radius of a rhodium atom in angstroms (A) and in meters $(\mathrm{m}) ?$ (b) How many Rh atoms would have to be placed side by side to span a distance of 6.0$\mu \mathrm{m}$ ? (c) If you assume that the Rh atom is a sphere, what is the volume in $\mathrm{m}^{3}$ of a single atom?

A golf ball has a diameter of 4.3 centimeters, and a tennis ball has a diameter of 6.9 centimeters. How much greater is the volume of the tennis ball?

Geometry What is the length of the edge of a cube if, after a slice 1 inch thick is cut from one side, the volume remaining

is 294 cubic inches?

Inflating a balloon The volume $V=(4 / 3) \pi r^{3}$ of a spherical

balloon changes with the radius.

a. At what rate $\left(\mathrm{ft}^{3} / \mathrm{ft}$ ) does the volume change with respect to \right.

the radius when $r=2 \mathrm{ft}$ ?

b. By approximately how much does the volume increase when

the radius changes from 2 to 2.2 $\mathrm{ft}$ ?

Find the volume of each sphere. Round to the nearest tenth.

The diameter is 12.5 centimeters.

A traditional unit of length in Japan is the ken $(1$ ken $=$ 1.97 $\mathrm{m} ) .$ What are the ratios of (a) square kens to square meters and (b) cubic kens to cubic meters? What is the volume of a cylindrical water tank of height 5.50 kens and radius 3.00 kens in (c) cubic kens and (d) cubic meters?

Volume A spherical snowball is melting. Find the approximate change in volume if the radius decreases from 3 $\mathrm{cm}$ to 2.8 $\mathrm{cm} .$

A jewelry bead is formed by drilling a $\frac{1}{2}-\mathrm{cm}$ radius hole from the center of a 1 -cm radius sphere. Explain why the volume is given by $\int_{1 / 2}^{1} 4 \pi x \sqrt{1-x^{2}} d x .$ Evaluate this integral or compute the volume in some easier way.

Games A billiard ball traverses a distance of 26 inches on a straight-line path, and then it collides with another ball, changes direction, and traverses a distance of 18 inches on a different straight-line path before coming to a stop. If an angle of $37^{\circ}$ is formed from the lines that connect the initial location of the ball to the final location of the ball and to the point of the collision, what are the two possible values of the distance $d$ between the initial and final locations of the ball? Sketch a figure first.

The driving distance for the top 100 golfers on the PGA tour is between 284.7 and 310.6

yards (Golfweek, March $29,2003 ) .$ Assume that the driving distance for these golfers is

uniformly distributed over this interval.

a. Give a mathematicalexpressionfor theprobability density function of driving distance.

b. What is the probability the driving distance for one of these golfers is less than 290 yards?

c. What is the probability the driving distance for one of these golfers is at least

300 yards?

d. What is the probability the driving distance for one of the se golfers is between 290

and 305 yards?

e. How many of these golfers drive the ball at least 290 yards?

Find the volume of a sphere with a circumference of $36 \pi \mathrm{ft}$.