Question

The radius of a circle is increasing at the ratio of \[0.1\text{ cm/s}\] .When the radius of the circle is \[\text{5 cm}\] , the rate of change of its area is :
A. \[\text{-}\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]
B. \[10\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]
C. \[0.1\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]
D. \[5\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]
E. \[\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]

Answer 1

Hint: Firstly we will look at the given terms in the question, then check what we have to find. After that, apply the formula of the area of the circle and then differentiate it with respect to \[t\] on both sides. Put the given values in the obtained differential to check which option is correct in the given options.

Complete step by step answer:
We know that a circle is a round shape and it has no corners. The interval joining a point to the centre is called radius.
The interval joining two points on the circle is called a chord and a chord that passes through the centre is called diameter.
You may have come across many objects in daily life which are rounded in shape like wheels, coins, button of shirts, dial of watch and many more. The collection of all the points in a plane which are at a fixed distance from a fixed point in the plane is called a circle.
In geometry you may define a circle as a closed, two dimensional curved shape.
The process of finding the differential coefficient of a function is called differentiation and the very important part of derivatives is found in its use in calculating the rate of change of quantities with respect to other quantities.
We know that \[\dfrac{dy}{dx}\] means the rate of change of \[y\] with respect to \[x\] hence \[\dfrac{dA}{dt}\] means rate of change of area with respect to time.


Now according to the question:
As we have given the radius of a circle is increasing at the ratio of \[0.1\text{ cm/s}\] hence, \[\dfrac{dr}{dt}=0.1\text{ cm/s}\]
The radius of the circle is \[\text{5 cm}\] hence, \[\text{r=5 cm}\]
Rate of change of its area is denoted by \[\dfrac{dA}{dt}\]
As we know that area of the circle is \[A=\pi {{r}^{2}}\]
Differentiating the area of circle with respect to \[t\] on both sides we will get:
\[\Rightarrow \]\[\dfrac{dA}{dt}=\dfrac{d}{dt}\pi {{r}^{2}}\]
\[\Rightarrow \]\[\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\]
Putting the values of \[r\] and \[\dfrac{dr}{dt}\] in the obtained differential we get:
\[\Rightarrow \]\[\dfrac{dA}{dt}=2\times \pi \times 5\times 0.1\]
\[\Rightarrow \]\[\dfrac{dA}{dt}=\pi \text{ c}{{\text{m}}^{\text{2}}}\text{/s}\]

So, the correct answer is “Option E”.

Note: If \[x\] is increasing as \[t\] increases then \[\dfrac{dx}{dt}\] is positive, but if \[x\] decreases as \[t\] increases then \[\dfrac{dx}{dt}\] is negative and same situation is for \[y\] also. Students must keep one thing in mind about a circle : that a circle can have infinite lines of symmetry and a circle can be divided equally irrespective of its size.