/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 The volume, \(V,\) of a cylindri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 13

The volume, \(V,\) of a cylindrical can is \(V=\pi r^{2} h\) and the total surface area, \(S\), of the can is $$ \begin{aligned} S &=\text { area of curved surface }+2 \cdot \text { (area of base) } \\ &=2 \pi r h+2 \pi r^{2} \end{aligned} $$ where \(r\) is the radius of the base and \(h\) is the height. a. Assume the height is three times the radius. Write the volume and the surface area as functions of the radius. b. As \(r\) increases, which grows faster, the volume or the surface area? Explain.

Short Answer

Expert verified
The volume, \( V(r) = 3 \pi r^{3} \), grows faster than the surface area, \( S(r) = 8 \pi r^{2} \), as \( r \) increases.

Step by step solution

01

Express the height in terms of the radius

Given that the height is three times the radius, we write: \( h = 3r \)
02

Substitute the height into the volume formula

The volume of the cylinder is given by \( V = \pi r^{2} h \). Substitute \( h = 3r \): \[ V = \pi r^{2} (3r) = 3 \pi r^{3} \] Therefore, the volume as a function of radius is \( V(r) = 3 \pi r^{3} \).
03

Substitute the height into the surface area formula

The surface area of the cylinder is given by: \[ S = 2 \pi r h + 2 \pi r^{2} \] Substitute \( h = 3r \): \[ S = 2 \pi r (3r) + 2 \pi r^{2} = 6 \pi r^{2} + 2 \pi r^{2} = 8 \pi r^{2} \] Therefore, the surface area as a function of radius is \( S(r) = 8 \pi r^{2} \).
04

Compare the growth rates of volume and surface area

To compare the growth rates, analyze how each expression depends on \( r \). The volume \( V(r) = 3 \pi r^{3} \) grows with \( r^{3} \), while the surface area \( S(r) = 8 \pi r^{2} \) grows with \( r^{2} \). Since \( r^{3} \) grows faster than \( r^{2} \), the volume grows faster than the surface area as \( r \) increases.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder Volume Formula
Now let鈥檚 compare how volume and surface area grow as the radius increases. To simplify: the volume formula V(r) = 3蟺r^3 depends on r^3, while the surface area formula S(r) = 8蟺r^2 depends on r^2.
This means:
* Volume increases by the cube of the radius. If the radius doubles, the volume multiplies by 2^3 = 8.
* Surface area increases by the square of the radius. If the radius doubles, the surface area multiplies by 2^2 = 4.
Since cubic growth (r^3) outpaces quadratic growth (r^2), volume grows faster than surface area as r increases.
So, while both increase, volume does so more rapidly. Knowing this difference helps with practical applications, like determining material costs or container sizing. This kind of growth comparison is essential in mathematics and real-life scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In "Love That Dirty Water" (Chicago Reader, April 5,1996 ), Scott Berinato interviewed Ernie Vanier, captain of the towboat Debris Control. The Captain said, "We've found a lot of bowling balls. You wouldn't think they'd float, but they do." When will a bowling ball float in water? The bowling rule book specifies that a regulation ball must have a circumference of exactly 27 inches. Recall that the circumference of a circle with radius \(r\) is \(2 \pi r\). The volume of a sphere with radius \(r\) is \(\frac{4}{3} \pi r^{3}\). a. What is a regulation bowling ball's radius in inches? b. What is the volume of a regulation bowling ball in cubic inches? (Retain at least two decimal places in your answer.) c. What is the weight in pounds of a volume of water equivalent in size to a regulation bowling ball? (Water weighs \(0.03612 \mathrm{lb} / \mathrm{in}^{3}\) ). d. A bowling ball will float when its weight is less than or equal to the weight of an equivalent volume of water. What is the heaviest weight of a regulation bowling ball that will float in water? e. Typical men's bowling balls are 15 or 16 pounds. Women commonly use 12 -pound bowling balls. What will happen to the men's and to the women's bowling balls when dropped into the water? Will they sink or float?

Using rules of logarithms, convert each equation to its power function equivalent in the form \(y=k x^{p}\). a. \(\log y=\log 4+2 \log x\) c. \(\log y=\log 1.25+4 \log x\) b. \(\log y=\log 2+4 \log x\) d. \(\log y=\log 0.5+3 \log x\)

(Graphing program optional.) Plot the functions \(f(x)=x^{2}\), \(g(x)=3 x^{2},\) and \(h(x)=\frac{1}{4} x^{2}\) on the same grid. Insert the symbol \(>\) or \(<\) to make the relation true. a. For \(x>0, g(x) \quad f(x) __________ h(x)\) b. For \(x<0, g(x) \quad f(x) __________ h(x)\)

Write a general formula to describe each variation. Use the information given to find the constant of proportionality. a. \(Q\) is directly proportional to both the cube root of \(t\) and the square of \(d\), and \(Q=18\) when \(t=8\) and \(d=3\). b. \(A\) is directly proportional to both \(h\) and the square of the radius, \(r\), and \(A=100 \pi\) when \(r=5\) and \(h=2\). c. \(V\) is directly proportional to \(B\) and \(h,\) and \(V=192\) when \(B=48\) and \(h=4\) d. \(T\) is directly proportional to both the square root of \(p\) and the square of \(u,\) and \(T=18\) when \(p=4\) and \(u=6\).

In parts (a)-(c), evaluate the functions at \(R\) and \(3 R\). a. \(C(r)=2 \pi r\), the circumference of a circle with radius \(r\) b. \(A(r)=\pi r^{2},\) the area of a circle with radius \(r\) c. \(V(r)=\frac{4}{3} \pi r^{3},\) the volume of a sphere with radius \(r\) d. Describe what happens to \(C(r), A(r)\), and \(V(r)\) when the radius triples from \(R\) to \(3 R\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.