#### To determine

**To find:** The value of dVdx at value of x=3 mm, if *V* is the volume of a cube with side length x, and explain its meaning.

#### Answer

The value of dVdx is 27mm3/mm_.

#### Explanation

Calculate the value of dVdx when x is 3mm and explain its meaning.

Use the below expression for volume.

V(x)=x3

Differentiate the volume equation.

dVdx=3x2 (1)

Substitute the value 3 for x in the above equation.

dVdx|x=3=3(3)2=27mm3/mm

The above value of 27mm3/mm is the rate at which the volume is increasing with the variable x.

Therefore, dVdx is 27mm3/mm.

#### To determine

**To show:** The rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube and explain geometrically why this result is true.

#### Explanation

Each cube will have 6 sides, and area of one side of the cube is x2.

Therefore, the surface area of the cube is as below.

s(x)=6x2

Refer the equation (1).

V'(x)=3x2

Simplify the equation as below.

V'(x)=126x2

Substitute s(x) for 6x2.

V'(x)=12s(x)

Thus, the rate of change of the volume of the cube is approximately half of its surface area.

Assume the condition in which a cube’s side length is *x* and the change in length of side is Δx on all sides.

Express the volume change, ΔV in terms of increased amount of length of side Δx.

ΔV=3x2(Δx)+3x(Δx)2+(Δx)3

The value of Δx is smaller and negligible.

Rewrite the above equation as below.

ΔV≃3x2ΔxΔVΔx≃3x2

Hence, the result is geometrically true.

Therefore, the rate of change of the volume of the cube is approximately half of its surface area and the result is geometrically true.