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Chapter 14: Q. 44 (page 897)

The surface area S of a sphere of radius r feet isS=S(r)=4蟺谤2. Find the instantaneous rate of change of the surface area with respect to the radius r atr=2.

Short Answer

Expert verified

If the Function for the surface area of a sphere is S=S(r)=4蟺谤2then the instantaneous rate of change of the surface area with respect to the radius r at2will be16蟺ft2ft

Step by step solution

01

Step 1. Given information 

Function for the surface area of a sphere is

S=S(r)=4蟺谤2

Radiusr=2

02

Step 2. The surface area of a sphere

Substitute r=2in the function for the surface area of a sphere

S(r)=4蟺谤2S(2)=4蟺(2)2 S(2)=16蟺

So surface area of the sphere at radiusr=2is16蟺ft2

03

Step 3. Instantaneous rate of change of surface area

Use instantaneous rate of change formula with respect r at 2

localid="1647033413018" S'(c)=limr鈫�cS(r)-S(c)r-cS'(2)=limr鈫�2S(r)-S(2)r-2S'(2)=limr鈫�24蟺谤2-16蟺r-2S'(2)=limr鈫�24蟺(r2-22)r-2S'(2)=limr鈫�24蟺(r+2)(r-2)r-2S'(2)=limr鈫�24蟺(r+2)S'(2)=4蟺(2+2)S'(2)=16蟺

So Instantaneous rate of change of surface area is16蟺ft2ft

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