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

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

b. See Explanation

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 = (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`