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

Hint:Apply the formula of the area of a circle that is, $A=\pi {{r}^{2}}$. Differentiate this area with respect to the radius and use the given information to find the value of $c$.
Complete step-by-step answer:
We have been given that when a circular plate is heated uniformly, its area expands $3c$ times as fast as its radius. This means that the change in area of the circular plate is $3c$ times that of change in the radius. Therefore, mathematically,
$dA=3cdr$, where $dA=$ change or expansion in the area of the circular plate and $dr=$ change or expansion in the radius of the circular plate.
Now, Let us come to the question.
We know that the area of a circle is ‘$\pi {{r}^{2}}$’. Therefore,
$A=\pi {{r}^{2}}$, now differentiating this area with respect to $r$ we get,
$\begin{aligned} & \Rightarrow \dfrac{dA}{dr}=\pi \times \dfrac{d{{r}^{2}}}{dr} \\\ & \Rightarrow \dfrac{dA}{dr}=\pi \times 2r \\\ & \Rightarrow \dfrac{dA}{dr}=2\pi r.....................(i) \\\ \end{aligned}$
Now, we have been given that $\dfrac{dA}{dr}=3c$ and we have to find the value of $c$ when $r=6$ units. Therefore, substituting these values in equation (i) we get,
$\begin{aligned} & 3c=2\pi \times 6 \\\ & c=\dfrac{12\pi }{3} \\\ & \therefore c=4\pi \\\ \end{aligned}$
Hence, option (a) is the correct answer.

Note: One may note that differentiation is the change of some function with respect to the variable on which it depends. We have applied the concept of derivative in the above question because in the question, the relation between the change of the function with respect to its variable is given. We cannot substitute the value of $r=6$ directly in the expression of the area of the circle.