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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 12E

a)

To determine

**To find:** The value of *V* is the volume of a cube with side length

Expert Solution

The value of

Calculate the value of

Use the below expression for volume.

Differentiate the volume equation.

Substitute the value

The above value of

Therefore,

b)

To determine

**To show:** The rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube and explain geometrically why this result is true.

Expert Solution

Each cube will have 6 sides, and area of one side of the cube is

Therefore, the surface area of the cube is as below.

Refer the equation (1).

Simplify the equation as below.

Substitute

Thus, the rate of change of the volume of the cube is approximately half of its surface area.

Assume the condition in which a cube’s side length is *x* and the change in length of side is

Express the volume change,

The value of

Rewrite the above equation as below.

Hence, the result is geometrically true.

Therefore, the rate of change of the volume of the cube is approximately half of its surface area and the result is geometrically true.