Question

Write a formula that estimates the change in the volume Vx of a cube when the edge lengths change from a to ax. Then use the formula to estimate the change in volume when x changes from cm to cm.

Answer 1

Answer:

a. $dV = 3x^2\ dx$

b. See Explanation

Step-by-step explanation:

Given

Shape: Cube

Solving (a); Formula that estimates the change in edge length

The volume (V) of a cube is:

$V = x^3$

Where

$x = edge\ length$

The change in volume is got by:

$dV = \frac{d}{dx}(x^3)$

Differentiate $x^3$

$dV = 3x^2\ dx$

Where

$dx = x_2 - x_1$ i.e. change in x

Solving (b):

The initial and final edge lengths are not given.

In order to solve this question, I'll assume that x changes from 5cm to 5.01cm

So, we have:

$x = x_1 = 5cm$

$x_2 = 5.01cm$

Substitute these values in $dV = 3x^2\ dx$

$dV = 3 * (5cm)^2 * (5.01cm - 5cm)$

$dV = 3 * (5cm)^2 * 0.01cm$

$dV = 3 * 25cm^2 * 0.01cm$

$dV = 075cm^3$