Question

Multiple Choice If the volume of a cube is increasing at 24 \(\mathrm{in}^{3} / \mathrm{min}\) and the surface area of the cube is increasing at 12 \(\mathrm{in}^{2} / \mathrm{min}\) , what is the length of each edge of the cube? \(\mathrm{}\) \(\begin{array}{lll}{\text { (A) } 2 \text { in. }} & {\text { (B) } 2 \sqrt{2} \text { in. (C) } \sqrt[3]{12} \text { in. (D) } 4 \text { in. }}\end{array}\)

Short answer

\((B) 2\sqrt{2} \mathrm{in}\)

Step-by-step solution

Translate the rates of change into equations

We can write the rates of change of volume and surface area as \(\frac{dV}{dt} = 3x^{2}\frac{dx}{dt} = 24\) and \(\frac{dA}{dt} = 12x\frac{dx}{dt} = 12\).

Solve the equations

Dividing the first equation by 3 and the second one by 12, we get \(x^{2}\frac{dx}{dt} = 8\) and \(x\frac{dx}{dt} = 1\). These equations tell us that the rate of change of the edge length with respect to time (\(\frac{dx}{dt}\)) is the same for both volume and surface area changes, and must be equal to 1/8. This results in \(\frac{dx}{dt} = \frac{1}{x}\) (from the equation for the surface area rate of change) and \(x^{2} = 8\) (from the equation for volume rate of change). Solving the second equation for x, we get \(x = \sqrt[2]{8} = 2\sqrt{2}\).

Validate the solution

Substitute the value of x obtained in step 2 into both the equations obtained in step 1. Check if both these equations hold true after substituting the value of x. If they hold true then the value of x is correct.