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Chapter 3: Q. 21 (page 299)

In Exercises 19鈥�26, write down an equation that relates the two quantities described. Then use implicit differentiation to obtain a relationship between the rates at which the quantities change over time.

The surface area S and height h of a cylinder with a fixed radius of 2 units.

Short Answer

Expert verified

The equation that relates the surface area S and the height of the cylinder h isS=4蟺丑+8蟺.

The derivative dSdtand dhdtare related by dSdt=4蟺dhdt.

Step by step solution

01

Step 1. Given information.

The radius of the cylinder is 2 units.

02

Step 2. Formula used.

The surface area of the cylinder is S=2蟺谤h+rsq. units.

03

Step 3. Apply the value of r.

Apply the value of r=2in S=2蟺谤h+ras follows.

S=2蟺谤h+rS=2蟺(2)(h+2)S=4蟺h+2)S=4蟺丑+8蟺

The equation that relates the surface area S and the height of the cylinderhisS=4蟺丑+8蟺.

04

Step 4. Apply the differentiation.

Apply the differentiation to S=4蟺丑+8蟺as follows.

role="math" localid="1648738419396" ddtS=ddt4蟺丑+8蟺dSdt=4蟺dhdt+0dSdt=4蟺dhdt

The derivativedSdtanddhdtare related bydSdt=4蟺dhdt.

05

Step 5. Conclusion.

The equation that relates the surface area S and the height of the cylinder h isS=4蟺丑+8蟺.

The derivativedSdtanddhdtare related bydSdt=4蟺dhdt.

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