Question

If the surface area of a cube is increasing at a rate of $5{\text{m}}^2\text{/sec},$ retaining its shape, then the rate of change in its volume (in ${\text{m}}^3\text{/sec}$), when the length of a side of the cube is $4\text{m},$ is

• A. $5.0$
• B. $2.5$
• C. $0.5$
• D. $1.5$

Answer 1

The correct option is A $5.0$

For a cube of side $a:S=6a^2\Rightarrow a=\sqrt{\left(\frac{S}{6}\right)}$
$V=a^3={\left(\frac{S}{6}\right)}^{\frac{3}{2}}$
$\Rightarrow \frac{dV}{dt}=\frac{1}{4}\cdot \sqrt{\frac{S}{6}}\cdot \frac{dS}{dt}$
When $a=4,S=96$
$\Rightarrow \frac{dV}{dt}=\frac{1}{4}\cdot \sqrt{16}\cdot 5=5$