Chapter 14: Q. 44 (page 897)

The surface area S of a sphere of radius r feet is $S = S (r) = 4 {πr}^{2}$ . Find the instantaneous rate of change of the surface area with respect to the radius r at $r = 2$ .

Short Answer

Expert verified

If the Function for the surface area of a sphere is $S = S (r) = 4 {πr}^{2}$ then the instantaneous rate of change of the surface area with respect to the radius r at $2$ will be $16 π \frac{{ft}^{2}}{ft}$

Step by step solution

Step 1. Given information

Function for the surface area of a sphere is

$S = S (r) = 4 {πr}^{2}$

Radius $r = 2$

Step 2. The surface area of a sphere

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

$S (r) = 4 {πr}^{2} S (2) = 4 π {(2)}^{2} S (2) = 16 π$

So surface area of the sphere at radius $r = 2$ is $16 π {ft}^{2}$

Step 3. Instantaneous rate of change of surface area

Use instantaneous rate of change formula with respect r at $2$

localid="1647033413018" $S' (c) = \lim_{r \to c} \frac{S (r) - S (c)}{r - c} S' (2) = \lim_{r \to 2} \frac{S (r) - S (2)}{r - 2} S' (2) = \lim_{r \to 2} \frac{4 {πr}^{2} - 16 π}{r - 2} S' (2) = \lim_{r \to 2} \frac{4 π (r^{2} - 2^{2})}{r - 2} S' (2) = \lim_{r \to 2} \frac{4 π (r + 2) (r - 2)}{r - 2} S' (2) = \lim_{r \to 2} 4 π (r + 2) S' (2) = 4 π (2 + 2) S' (2) = 16 π$

So Instantaneous rate of change of surface area is $16 π \frac{{ft}^{2}}{ft}$

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The surface area S of a sphere of radius r feet is $S = S (r) = 4 {πr}^{2}$ . Find the instantaneous rate of change of the surface area with respect to the radius r at $r = 2$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The surface area of a sphere

Step 3. Instantaneous rate of change of surface area

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The surface area S of a sphere of radius r feet isS=S(r)=4πr2. Find the instantaneous rate of change of the surface area with respect to the radius r atr=2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The surface area of a sphere

Step 3. Instantaneous rate of change of surface area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Pure Maths

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

The surface area S of a sphere of radius r feet is $S = S (r) = 4 {πr}^{2}$ . Find the instantaneous rate of change of the surface area with respect to the radius r at $r = 2$ .