Question

The side of a cube is equal to the radius of a sphere. If the sides and the radius increases at the same rate, then the ratio of the rate of change in surface area of the sphere to that of cube is

• A. greater than $3$
• B. less than $2$ but greater than $1$
• C. less than $1$
• D. greater than $2$ but less than $3$

Answer 1

The correct option is D greater than $2$ but less than $3$

Let side of a cube is $x$, radius of sphere is $r$.
Given, $x=r$ and $\frac{dx}{dt}=\frac{dr}{dt}$
Let $S_1,S_2$ be the surface areas of cube and sphere respectively.
$\Rightarrow S_1=6x^2$
$\Rightarrow \frac{d{S}_1}{dt}=12x\frac{dx}{dt}\cdots \left(1\right)$ and
$S_2=4\pi r^2\Rightarrow \frac{d{S}_2}{dt}=8\pi r\frac{dr}{dt}\cdots \left(2\right)$
Dividing $\left(2\right)$ by $\left(1\right)$ we have the required ratio $=\frac{8\pi }{12}$ which is clearly greater than $2$ but less than $3$