Q. 17 Suppose the radius r, height h, ... [FREE SOLUTION]

Chapter 3: Q. 17 (page 299)

Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further suppose that the height of the cylinder is always twice its radius. Write $\frac{d V}{d t}$ in terms of h and $\frac{d h}{d t}$ .

Short Answer

Expert verified

The derivative $\frac{d V}{d t}$ in terms ofhand $\frac{d h}{d t}$ is $\frac{dV}{dt} = \frac{3}{4} {蟺丑}^{2} \frac{dh}{dt}$ .

Step by step solution

Step 1. Given information.

Given that the radius r, height h, volume V of a cylinder are functions of time t, and the height of the cylinder is always twice its radius.

That is, the height is $h = 2 r$ .

From $h = 2 r$ , the radius is $r = \frac{h}{2}$ .

Step 2. Formula used.

The volume V of the cylinder is given by the formula $V = {蟺谤}^{2} h$ cu. units.

Step 3. Apply the value of r.

Apply the value $r = \frac{h}{2}$ in $V = {蟺谤}^{2} h$ as follows.

$V = {蟺谤}^{2} h V = 蟺 {(\frac{h}{2})}^{2} h V = \frac{1}{4} {蟺丑}^{3}$

Step 4. Apply the differentiation.

Apply the differentiation to $V = \frac{1}{4} {蟺丑}^{3}$ with respect to tas follows.

$\frac{d}{dt} (V) = \frac{d}{d t} (\frac{1}{4} {蟺丑}^{3}) \frac{d V}{d t} = \frac{1}{4} 蟺 (3 h^{2} \frac{dh}{dt}) \frac{dV}{dt} = \frac{3}{4} {蟺丑}^{2} \frac{dh}{dt}$

Step 5. Conclusion.

The derivative $\frac{d V}{d t}$ in terms of hand $\frac{d h}{d t}$ is $\frac{dV}{dt} = \frac{3}{4} {蟺丑}^{2} \frac{dh}{dt}$ .

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91影视

Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further suppose that the height of the cylinder is always twice its radius. Write $\frac{d V}{d t}$ in terms of h and $\frac{d h}{d t}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Apply the value of r.

Step 4. Apply the differentiation.

Step 5. Conclusion.

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91影视

Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further suppose that the height of the cylinder is always twice its radius. Write dVdtin terms of h and dhdt.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Apply the value of r.

Step 4. Apply the differentiation.

Step 5. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Pure Maths

Discrete Mathematics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further suppose that the height of the cylinder is always twice its radius. Write $\frac{d V}{d t}$ in terms of h and $\frac{d h}{d t}$ .