Chapter 3: Q 39. (page 299)

Suppose the sides of a cube are expanding at a rate of $2$ inches per minute.
How fast is the volume of the cube changing at the moment that the cube has a side length of $8$ inches?

Short Answer

Expert verified

The rate of change in volume is $384 i n^{3} / m i n$

Step by step solution

Step 1. Given Information

It is given that

$\frac{d s}{d t} = 2 i n / m i n, s = 8 i n$

Step 2. Setting up the rate of change of the volume equation

The formula for the volume of a cube is

$V = s^{3}$

where

V is the volume of a cube

s is the side length

Find the derivative with respect to t

$\frac{d V}{d t} = 3 s^{2} \frac{d s}{d t}$

Step 3. Finding the rate of change in Volume

Plug values $\frac{d s}{d t} = 2 i n / m i n, s = 8 i n$ into volume derivative

$\frac{d V}{d t} = 3 s^{2} \frac{d s}{d t} \frac{d V}{d t} = 3 {(8)}^{2} \times 2 \frac{d V}{d t} = 384 i n^{3} / i n$

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Suppose the sides of a cube are expanding at a rate of $2$ inches per minute.
How fast is the volume of the cube changing at the moment that the cube has a side length of $8$ inches?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Setting up the rate of change of the volume equation

Step 3. Finding the rate of change in Volume

One App. One Place for Learning.

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Suppose the sides of a cube are expanding at a rate of 2inches per minute.How fast is the volume of the cube changing at the moment that the cube has a side length of 8inches?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Setting up the rate of change of the volume equation

Step 3. Finding the rate of change in Volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Pure Maths

Discrete Mathematics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Suppose the sides of a cube are expanding at a rate of $2$ inches per minute.
How fast is the volume of the cube changing at the moment that the cube has a side length of $8$ inches?