Question

If a circular plate is heated uniformly, its area expands $3c$ times as fast as its radius, then the value of $c$ when the radius is $6$ units, is

• A. $4\pi$
• B. $2\pi$
• C. $6\pi$
• D. $3\pi$
• E. $8\pi$

Answer 1

The correct option is C $6\pi$

Let area of the circular plate be $A$ and its radius be $r$.
Then, $A=\pi r^2$
On differentiating both sides with respect to $t$, we get
$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$
But $\frac{dA}{dt}=3c\frac{dr}{dt}$
Therefore, $3c\frac{dr}{dt}=2\pi r\frac{dr}{dt}$
$\Rightarrow 3c=2\pi r$
On putting $r=6$, we get $c=4\pi$