Answer:
a. [tex]dV = 3x^2\ dx[/tex]
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:
[tex]V = x^3[/tex]
Where
[tex]x = edge\ length[/tex]
The change in volume is got by:
[tex]dV = \frac{d}{dx}(x^3)[/tex]
Differentiate [tex]x^3[/tex]
[tex]dV = 3x^2\ dx[/tex]
Where
[tex]dx = x_2 - x_1[/tex] 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:
[tex]x = x_1 = 5cm[/tex]
[tex]x_2 = 5.01cm[/tex]
Substitute these values in [tex]dV = 3x^2\ dx[/tex]
[tex]dV = 3 * (5cm)^2 * (5.01cm - 5cm)[/tex]
[tex]dV = 3 * (5cm)^2 * 0.01cm[/tex]
[tex]dV = 3 * 25cm^2 * 0.01cm[/tex]
[tex]dV = 075cm^3[/tex]
