Question

The radius of a circular plate is increasing at the ratio of 0.20 cm/sec. At what rate is the area increasing when the radius of the plate is 25 cm.

Answer 1

Hint: Write the formula for the area of a circle given as \[\text{Area}=A=\pi {{r}^{2}},\] where r is the radius of the circle. Now, differentiate the area relation with respect to time and substitute \[\dfrac{dr}{dt}=0.20cm/\sec .\] Finally, substitute r = 25 cm in the obtained differential equation and get the value of \[\dfrac{dA}{dt}.\]

Complete step by step answer:
We have been given that the radius of the circular plate is increasing at the ratio of 0.20 cm/sec. We have to determine the rate of the increase of area at the time when the radius of the plate becomes 25 cm. Now, we know that the area of a circle is given as \[\text{Area}=A=\pi {{r}^{2}},\] where ‘A’ is the area and ‘r’ is the radius of the circle. Therefore, on differentiating this area relation with respect to time (t), we get,
\[\Rightarrow \dfrac{dA}{dt}=\dfrac{d\left[ \pi {{r}^{2}} \right]}{dt}\]
Using the chain rule of differentiation,
\[\Rightarrow \dfrac{dA}{dt}=\pi \left[ \dfrac{d{{r}^{2}}}{dr}\times \dfrac{dr}{dt} \right]\]
\[\Rightarrow \dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\]
Substituting the given value of \[\dfrac{dr}{dt}=0.20cm/\sec ,\] we get,
\[\Rightarrow \dfrac{dA}{dt}=2\pi r\times 0.20\]
\[\Rightarrow \dfrac{dA}{dt}=0.4\pi r.....\left( i \right)\]
Now, we have to find the value of \[\dfrac{dA}{dt}\] when the radius of the plate is 25 cm. So, substituting r = 25 cm in equation (i), we get,
\[\Rightarrow \dfrac{dA}{dt}=0.4\pi \times 25\]
Substituting \[\pi =3.14,\] we get,
\[\Rightarrow \dfrac{dA}{dt}=0.4\times 3.14\times 25\]
\[\Rightarrow \dfrac{dA}{dt}=31.4c{{m}^{2}}/\sec \]
Hence, the rate of the increase of the area of the circular plate is \[31.4c{{m}^{2}}/\sec .\]

Note:
One may note that the process of differentiation denotes the change of some function with respect to the variable on which it depends. We have applied the concept of the derivative in the above question because in the question the rate of change of radius was given and we were required to find the rate of change of area. We cannot substitute the value of r = 25cm directly in the expression of the area of the circle as it will be a wrong approach. You may note that we have substituted \[\pi =3.14\] in the solution because there was no information provided in the question. You may also use \[\pi =\dfrac{22}{7}.\] At last, remember that the unit of rate of change of area will be \[c{{m}^{2}}/\sec,\] so do not forget to mention it otherwise the answer will be considered dimensionally wrong.