Question

The radius of a sphere is changing at the rate of 0.1 cm/s. The rate of change of its surface area when the radius is 200 cm, is \[\]
A. $ 8\pi \text{ c}{{\text{m}}^{2}}/\text{sec} $ \[\]
B. $ 12\pi \text{ c}{{\text{m}}^{2}}/\text{sec} $ \[\]
C. $ 160\pi \text{ c}{{\text{m}}^{2}}/\text{sec} $ \[\]
D. $ 200\pi \text{ c}{{\text{m}}^{2}}/\text{sec} $ \[\]

Answer 1

Hint: We recall the surface area of a sphere with radius $ r $ is given by $ A=4\pi {{r}^{2}} $ . We use the idea that the rate of change of any measurable quantity can be expressed as derivative of that quantity with the time variable $ t $ . We differentiate $ A=4\pi {{r}^{2}} $ with respect to $ t $ and put $ r,\dfrac{dr}{dt} $ which are given in the question to get the answer. \[\]

Complete step by step answer:
We know that a sphere is the locus of the outline of a circle when it is moved one turn around its fixed diameter. The surface area of sphere is only curved and the total surface area of the sphere is given by
\[A=4\pi {{r}^{2}}\]
We also know that the rate of change of any measurable quantity is given by the derivative of the quantity with respect to time as an independent variable. We denote the time variable as $ t $ and differentiate the surface area with respect to $ t $ and have;
\[\begin{align}
  & \dfrac{d}{dt}A=\dfrac{d}{dt}\left( 4\pi {{r}^{2}} \right) \\
 & \Rightarrow \dfrac{dA}{dt}=4\pi \dfrac{d}{dt}\left( {{r}^{2}} \right) \\
\end{align}\]
We use the standard differentiation $ \dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}} $ to have;
\[\begin{align}
  & \Rightarrow \dfrac{dA}{dt}=4\pi \cdot 2r\dfrac{dr}{dt} \\
 & \Rightarrow \dfrac{dA}{dt}=8\pi r\dfrac{dr}{dt} \\
\end{align}\]

We are given in the question that the radius of a sphere is changing at the rate of 0.1 cm/s which means $ \dfrac{d}{dt}r=\dfrac{dr}{dt}=0.1\text{ cm/sec} $ . We are asked to find the rate of change of its surface area when the radius is 200 cm which means at $ r=200 $ cm. We put $ r=200,\dfrac{dr}{dt}=0.1 $ in the above step to have;
\[\Rightarrow \dfrac{dA}{dt}=8\pi \times 200\times 0.1=160\pi \text{ c}{{\text{m}}^{2}}/s\]
Here $ \dfrac{dA}{dt} $ represents rate of change of surface area and the correct choice is C. \[\]


Note:
 We must properly place the units of rate of change. We see that the radius is a function of time here $ r\left( t \right) $ and its rate of change is constant and is also increasing. The rate of change with derivative is called instantaneous rate of change and without derivative is called average rate of change which is given by $ \dfrac{f\left( {{t}_{2}} \right)-f\left( {{t}_{1}} \right)}{{{t}_{2}}-{{t}_{1}}} $ .