Answered

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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.

Sagot :

MrRoyal

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]

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