Question

If surface area of a cube is changing at a rate of $5m^2/s$ , find the rate of change of body diagonal at the moment when side length is $1m$.

• A. $5m/s$
• B. $5\sqrt{3}m/s$
• C. $\frac{5}{2}\sqrt{3}m/s$
• D. $\frac{5}{4\sqrt{3}}m/s$

Answer 1

The correct option is D $\frac{5}{4\sqrt{3}}m/s$

Surface area of cube $S=6a^2$
(where, $a$ = side of cube)
Body diagonal $l=\sqrt{3}a$.
$\frac{dS}{dt}=5m/s$

Therefore $S=2l^2$
Differentiating $S$ w.r.t. time,
$\frac{dS}{dt}=2\left(2l\right)\frac{dl}{dt}\Rightarrow dl/dt=\frac{1}{4\left(\sqrt{3}a\right)}.\frac{dS}{dt}\frac{dS}{dt}=\frac{5}{4\left(\sqrt{3}a\right)}m/s$