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Chapter 3: Q. 13 (page 299)

Suppose the radius $r$ , volume $V$ , and surface area $S$ of a sphere are functions of time $t$ .
(a) How are $\frac{d V}{d t}$ and $\frac{d r}{d t}$ related?
(b) How are $\frac{d S}{d t}$ and $\frac{d r}{d t}$ related?

Short Answer

Expert verified

(a)The derivative $\frac{d V}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{d V}{d t} = 4 {蟺谤}^{2} \frac{dr}{dt}$ .

(b)The derivative $\frac{d S}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{d S}{d t} = 8 蟺谤 \frac{dr}{dt}$ .

Step by step solution

01

Part (a) Step 1. Formula used.

The volume V of the sphere is given by the formula

$V = \frac{4}{3} {蟺谤}^{3}$ cu. units.

02

Part (a) Step 2. Apply the differentiation.

Since V and r are functions of t.

Apply the differentiation to $V = \frac{4}{3} {蟺谤}^{3}$ with respect to tas follows.

$\frac{d V}{d t} = \frac{4}{3} 蟺 (3 r^{2} \frac{dr}{dt}) \frac{dV}{dt} = 4 {蟺谤}^{2} \frac{dr}{dt}$

03

Part (a) Step 3. Conclusion

The derivatives $\frac{d V}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{dV}{dt} = 4 {蟺谤}^{2} \frac{dr}{dt}$ .

04

Part (b) Step 1. Formula used.

The surface area of the sphere is given by the formula $S = 4 {蟺谤}^{2}$ sq. units.

05

Part (b) Step 2. Apply the differentiation.

Given S and r are functions of t.

Apply the differentiation to $S = 4 {蟺谤}^{2}$ with respect to t as follows.

$\frac{d S}{d t} = 4 蟺 (2 r \frac{dr}{dt}) \frac{dS}{dt} = 8 蟺谤 \frac{dr}{dt}$

06

Part (b) Step 3. Conclusion.

The derivatives $\frac{d S}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{dS}{dt} = 8 蟺谤 \frac{dr}{dt}$ .

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Most popular questions from this chapter

Determine whether or not each function f in Exercises 53鈥�60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

$f (x) = \ln (x^{2} + 1), [a, b] = [0, 1]$

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = \cos (3 (x - \frac{蟺}{2}))$

Find the possibility graph of its derivative f'.

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = x^{3} + 3 x^{2}$

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

See all solutions

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