Question

If surface area of a cube is changing at a rate of $18m^2/s$ , find the rate at which length of the body diagonal changes, at a moment when side length is $3m$.

• A. $18\text{m/s}$
• B. $3\sqrt{3}\text{m/s}$
• C. $\frac{3}{2}\text{m/s}$
• D. $\frac{\sqrt{3}}{2}\text{m/s}$

Answer 1

The correct option is D $\frac{\sqrt{3}}{2}\text{m/s}$

Surface area of a cube is
$S=6a^2$ (where $a$ = side of cube)

Body diagonal length $l$ is
$l=\sqrt{3}a$

$\therefore S=2l^2$

Differentiating $S$ w.r.t. time,
$\frac{dS}{dt}=2\left(2l\right)\frac{dl}{dt}$
$\Rightarrow \frac{dl}{dt}=\frac{1}{4\left(\sqrt{3}a\right)}\frac{dS}{dt}(\because \frac{dS}{dt}=18\frac{m^2}{s},a=3m)$

$\Rightarrow \frac{dl}{dt}=\frac{18}{12\sqrt{3}}=\frac{\sqrt{3}}{2}\text{m/s}$