Question

An edge of a variable cube is increasing at a rate of $10cm/sec$. How fast the volume of the cube will increase when the edge is $5cm$ long.

• A. $750c{m}^3/sec$
• B. $75c{m}^3/sec$
• C. $300c{m}^3/sec$
• D. $150c{m}^3/sec$
• E. $25c{m}^3/sec$

Answer 1

The correct option is A

$750c{m}^3/sec$

Step 1. Given data:

The variable edge of the cube, $\frac{dx}{dt}=10cm/sec$

The edge of the cube, $x=5cm$

Step 2. Formula used:

The volume of the cube, $V=x^3.....\left(1\right)$

Step 3. Finding the increase in the volume of the cube:

Differentiating both sides of the equation $\left(1\right)$, we get

$$\begin{gathered} \frac{dV}{dt}=3x^2\left(\frac{dx}{dt}\right) \\ \frac{dV}{dt}=3{\left(5\right)}^2\left(10\right) \\ \frac{dV}{dt}=750c{m}^3/sec \end{gathered}$$

Hence, the volume of the cube is increasing at the rate of $750c{m}^3/sec$

Hence, option A is correct.