   Chapter 2.7, Problem 15E

Chapter
Section
Textbook Problem

# A spherical balloon is being inflated. Find the rate of increase of the surface area ( S =   4 π r 2 ) with respect to the radius r when r is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make?

To determine

To find: The rate of increase of the surface area with respect to the radius r when r is

(a) 1 ft, (b) 2 ft, and (c) 3 ft then make conclusion.

Explanation

Conclusion:

We can conclude that, the rate of increase of surface area of balloon is linear with respect to the radius..

1) Concept:

Use concept of derivative to find the rate of increase of the surface area with respect to the radius r.

2) Formula:

i. Surface area of a spherical balloon  S= 4πr2

ii. Power rule of derivative:

ddxxn=nxn-1

3) Given:

S= 4πr2

(a) r = 1 ft, (b) r = 2 ft, and (c) r = 3 ft

4) Calculation:

Surface area of a spherical balloon S= 4πr2

Now, find derivative of S= 4πr2 with respect to radius r.

By using power rule of derivative,

ddrS=ddr(4πr2)

dSdr=8πr

(a) To find the rate of increase of the surface area at r = 1, substitute r = 1 in dSdr=8πr

dSdr=8

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