Chapter 3: Q. 15 (page 299)

Suppose the radius r, height h, and volume V of a cylinder are functions of time t. How is $\frac{d V}{d t}$ related to $\frac{d r}{d t}$ if the height of the cylinder is constant?

Short Answer

Expert verified

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

Step by step solution

Step 1. Formula used.

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

Step 2. Apply the differentiation.

Given that the radius r, height h, and volume V are functions oft.

Apply the differentiation to $V = {πr}^{2} h$ with respect to t and keeping h as constant as follows.

role="math" localid="1648728564870" $\frac{d V}{d t} = 2 πrh \frac{dr}{dt}$

Step 3. Conclusion.

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

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Suppose the radius r, height h, and volume V of a cylinder are functions of time t. How is $\frac{d V}{d t}$ related to $\frac{d r}{d t}$ if the height of the cylinder is constant?

Short Answer

Step by step solution

Step 1. Formula used.

Step 2. Apply the differentiation.

Step 3. Conclusion.

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Step by step solution

Step 1. Formula used.

Step 2. Apply the differentiation.

Step 3. Conclusion.

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Suppose the radius r, height h, and volume V of a cylinder are functions of time t. How is $\frac{d V}{d t}$ related to $\frac{d r}{d t}$ if the height of the cylinder is constant?