Question

The surface area of a cube increases at a uniform rate of $0.5\text{cm}{}^2\text{/sec}$. The rate of increase in the volume of the cube when the side is $20$ cm is ____

• A. $1.25\text{cm}{}^3\text{/sec}$
• B. $5\text{cm}{}^3\text{/sec}$
• C. $2.5\text{cm}{}^3\text{/sec}$
• D. $0.75\text{cm}{}^3\text{/sec}$

Answer 1

The correct option is C

$2.5\text{cm}{}^3\text{/sec}$

$s=6x^2$

$\frac{ds}{dt}=12x\frac{dx}{dt}=\frac{1}{2}$

$v=x^3$

$\frac{dv}{dt}=3x^2\frac{dx}{dt}$

$=3x^2\left(\frac{1}{24x}\right)$

$=3\times 20\times \frac{1}{24}=2.5\text{cm}{}^3\text{/sec}$