Question

If metallic circular plate of radius $50{\text{cm}}$ is heated so that its radius increases at the rate of $1{\text{mm}}$ per hour, then the rate at which the area of the plate increases (in ${\text{c}}{{\text{m}}^2}/{\text{hr}}$) is
A) $5\pi $
B) $10\pi $
C) $100\pi $
D) $50\pi $

Answer 1

Hint:
Here, we will use the formula of area of the circle and differentiate it with respect to radius $r$. Then, substituting the value of the radius as well as the increase in the radius will help us to find the required rate at which the area of the plate increases.

Formula Used: We will use the following formulas:
1) Area of a circle, $A = \pi {r^2}$, where $r$ is the radius of the circle.
2) $\dfrac{{dy}}{{dx}}{x^n} = n{x^{n - 1}}$

Complete step by step solution:
According to the question, the radius of a metallic circular plate, $r = 50{\text{cm}}$
Now, we know the area of a circle, $A = \pi {r^2}$, where $r$ is the radius of the circle.
Now, differentiating both the sides with respect to $r$ using the formula, $\dfrac{{dy}}{{dx}}{x^n} = n{x^{n - 1}}$, we get
$\dfrac{{dA}}{{dr}} = 2\pi r$
Hence, this can also be written as:
$dA = 2\pi r \times dr$…………………………………….$\left( 1 \right)$
Now, as we know, $r = 50{\text{cm}}$
Also, it is given that:
When the metallic circular plate is heated, its radius increases at the rate of $1{\text{mm}}$per hour
So, \[dr = 1{\text{mm}}/{\text{hr}}\]
Now, we know that $1{\text{cm}} = 10{\text{mm}}$
Hence, dividing both sides by 10, we get,
$\dfrac{1}{{10}}{\text{cm}} = 1{\text{mm}}$
Therefore, $dr = \dfrac{1}{{10}}{\text{cm}}/{\text{hr}}$
Hence, now the units of the radius and the increase in radius are the same.
Substituting these values in $\left( 1 \right)$, we get,
$dA = 2\pi \left( {50} \right) \times \dfrac{1}{{10}}{\text{c}}{{\text{m}}^2}/{\text{hr}}$
\[ \Rightarrow dA = \left( {100\pi \times \dfrac{1}{{10}}} \right) = 10\pi {\text{c}}{{\text{m}}^2}/{\text{hr}}\]
Therefore, the rate at which the area of the plate increases (in ${\text{c}}{{\text{m}}^2}/{\text{hr}}$) is \[10\pi {\text{c}}{{\text{m}}^2}/{\text{hr}}\]
This is because $dA$ represents the change in area.

Hence, option B is the correct answer.

Note:
In calculus differentiation is a method of finding the derivative of a function. It is a process in which we find the instantaneous rate of change in function based on one of its variables. The opposite of finding a derivative is anti-differentiation also known as integration.
Now, for example, if $x$ is a variable and $y$ is another variable, then the rate of change of $x$ with respect to $y$ is given by $\dfrac{{dy}}{{dx}}$. This is the general expression of derivative of a function and is represented as $f'\left( x \right) = \dfrac{{dy}}{{dx}}$, where $y = f\left( x \right)$ is any given function.