Question

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 4 mm. V'(4) = mm3/mm What does V'(4) mean in this situation?

Answer 1

Answer:

V'(4) = 48 mm³/mm

V'(4) means that the rate of change of volume with respect to side when the length of the side is 4 mm

Step-by-step explanation:

Data provided in the question

V is the volume of the cube with x as its sides

Also,

Volume of cube = side³

thus,

V = x³ mm³

Now,

Derivative of volume with respect to side x,

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

or

$\frac{\textup{dv}}{\textup{dx}}$  = V'(x) = 3x²

now,

at x = 4 mm

V'(4) = 3(4)²

or

V'(4) = 48 mm³/mm

Here, V'(4) means that the rate of change of volume with respect to side when the length of the side is 4 mm

Answer 2

The volume of a cube is the amount of space in the cube.

• The value of V'(4) is 48
• The rate of change of the volume when the side length of the cube is 4mm, is 48mm^3/mm

The volume of a cube is:

$\mathbf{V(x) = x^3}$

Where: x represents the measure of the sides of the cube

Differentiate with respect to x

$\mathbf{V'(x) = 3x^2}$

Substitute 4 for x

$\mathbf{V'(4) = 3 \times 4^2}$

$\mathbf{V'(4) = 48}$

The value of V'(4) is 48

The interpretation is that:

The rate of change of the volume when the side length of the cube is 4mm, is 48mm^3/mm

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