Q. 45 The volume聽V of a sphere of rad... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 14: Q. 45 (page 897)

The volumeV of a sphere of radius r feet is $V = V (r) = \frac{4}{3} {蟺谤}^{3}$ . Find the instantaneous rate of change of the volume with respect to the radius r at $r = 2 .$

Short Answer

Expert verified

If the function for the volume of a sphere is $V = V (r) = \frac{4}{3} {蟺谤}^{3}$ then the instantaneous rate of change of the volume with respect to the radius r at $r = 2$ will be $16 蟺 \frac{{ft}^{3}}{ft} .$

Step by step solution

Step 1. Given information

Function for the volume of a sphere is

$V = V (r) = \frac{4}{3} {蟺谤}^{3}$

Radius $r = 2$

Step 2. The Volume of a sphere

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

$V (r) = \frac{4}{3} {蟺谤}^{3} V (2) = \frac{4}{3} 蟺 {(3)}^{3} V (2) = \frac{4 脳 9}{3} 蟺 V (2) = 12 蟺$

So the volume of the sphere at the radius $r = 2$ is $12 蟺 {ft}^{3}$

Step 3. Instantaneous rate of change of volume

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

localid="1647033372487" $V' (c) = \lim_{r 鈫� c} \frac{V (r) - V (c)}{r - c} V' (2) = \lim_{r 鈫� 2} \frac{V (r) - V (2)}{r - 2} V' (2) = \lim_{r 鈫� 2} \frac{\frac{4}{3} {蟺谤}^{3} - 12 蟺}{r - 2} V' (2) = \lim_{r 鈫� 2} \frac{\frac{4}{3} 蟺 (r^{3} - 8)}{r - 2} V' (2) = \lim_{r 鈫� 2} \frac{\frac{4}{3} 蟺 (r^{2} + 2 r + 4) (r - 2)}{r - 2} V' (2) = \lim_{r 鈫� 2} \frac{4}{3} 蟺 (r^{2} + 2 r + 4) V' (2) = \frac{4}{3} 蟺 (2^{2} + 2 (2) + 4) V' (2) = 16 蟺$

So Instantaneous rate of change of volume islocalid="1647032540665" $16 蟺 \frac{{ft}^{3}}{ft}$

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91影视

The volumeV of a sphere of radius r feet is $V = V (r) = \frac{4}{3} {蟺谤}^{3}$ . Find the instantaneous rate of change of the volume with respect to the radius r at $r = 2 .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. The Volume of a sphere

Step 3. Instantaneous rate of change of volume

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91影视

The volumeV of a sphere of radius r feet isV=V(r)=43蟺谤3. Find the instantaneous rate of change of the volume with respect to the radius r atr=2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The Volume of a sphere

Step 3. Instantaneous rate of change of volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

The volumeV of a sphere of radius r feet is $V = V (r) = \frac{4}{3} {蟺谤}^{3}$ . Find the instantaneous rate of change of the volume with respect to the radius r at $r = 2 .$