Question

The radius of the circular plate is 2 cm. If it is being heated the expansion in its radius happens at a rate of $0.02cm/\sec $. Find the rate of expansion of the area when the radius is 2.1 cm.

Answer 1

Hint: The radius and the area of the plate both expands when it’s being heated. We find the relation between the radius and the area of the plate. We differentiate both sides to find the rate of expansion of an area dependent on the rate of expansion of radius. We place the values to find a solution to the problem.

Complete step-by-step solution
The radius of the circular plate is 2 cm. If the plate is being heated then both the radius and the area of the plate get expanded.
We assume the radius of the plate as “r” and the area of the plate as A. We know that the area of a circular plate with radius r is $A=\pi {{r}^{2}}............(i)$.
On being heated the expansion rate in its radius happens at a rate of $0.02cm/\sec $.
We need to find the rate of expansion of the area when the radius is 2.1 cm. So, we put $r=2.1$.
We express the expansion rate in its radius and the area as $\dfrac{dr}{dt}$ and $\dfrac{dA}{dt}$ respectively.
Here, t denotes the time. We have been given $\dfrac{dr}{dt}=0.02cm/\sec $.
Now we differentiate the given equation (i) with respect to t.
$\begin{align}
  & A=\pi {{r}^{2}} \\
 & \Rightarrow \dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt} \\
\end{align}$
To find the value of $\dfrac{dA}{dt}$, we put the rest of the values.
So, $\dfrac{dA}{dt}=\dfrac{2\times 22\times 2.1\times 0.02}{7}=0.262c{{m}^{2}}/\sec $.
The rate of expansion of area having the plate’s radius of 2.1 cm is $0.262c{{m}^{2}}/\sec $.

Note: The unit in case of $\dfrac{dA}{dt}$ depends on the unit multiplication of radius and rate of expansion of radius. We are finding the instantaneous rate and that’s why we don’t need to use the primary radius which was 2 cm before heating started.