Question

If at any instant $t$, for a sphere, $r$ denotes the radius, $S$ denotes the surface area and $V$ denotes the volume, then what is $\dfrac{{dV}}{{dt}}$?
A. $\dfrac{1}{2}S\dfrac{{dr}}{{dt}}$
B. $\dfrac{1}{2}r\dfrac{{dS}}{{dt}}$
C. $r\dfrac{{dS}}{{dt}}$
D. $\dfrac{1}{2}{r^2}\dfrac{{dS}}{{dt}}$

Answer 1

Hint: Given, at any instant $t$, for a sphere $r$ be the radius of the sphere, $S$ denotes the surface area of the sphere and $V$ denotes the volume of the sphere. We have to find $\dfrac{{dV}}{{dt}}$ . First, we use the surface area of the sphere and then differentiate it with respect to $t$. Then we use the volume of the sphere then differentiate volume with respect to $t$.

Formula Used:
Surface area of sphere $S = 4\pi {r^2}$
Volume of sphere $V = \dfrac{4}{3}\pi {r^3}$

Complete step by step Solution:
For finding the derivative of a function we use differentiation. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables. 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 $f'(x) = \dfrac{{dy}}{{dx}}$, where $y = f(x)$ is any function.
Given, at any instant $t$, for a sphere $r$ be the radius of the sphere, $S$ denotes the surface area of the sphere and $V$ denotes the volume of the sphere.
The surface area of a sphere
$S = 4\pi {r^2}$
Differentiating with respect to $t$
$\dfrac{{dS}}{{dt}} = 4\pi \times 2r \times \dfrac{{dr}}{{dt}}$
$ \Rightarrow \dfrac{{dS}}{{dt}} = 8\pi r\dfrac{{dr}}{{dt}}$
Finding the value of $\dfrac{{dr}}{{dt}}$
$\dfrac{{dr}}{{dt}} = \dfrac{1}{{8\pi r}}\dfrac{{dS}}{{dt}}$
Volume of sphere
$V = \dfrac{4}{3}\pi {r^3}$
Differentiating with respect to $t$
$\dfrac{{dV}}{{dt}} = \dfrac{4}{3}\pi \times 3{r^2} \times \dfrac{{dr}}{{dt}}$
$ \Rightarrow \dfrac{{dV}}{{dt}} = 4\pi {r^2}\dfrac{{dr}}{{dt}}$
Putting value of $\dfrac{{dr}}{{dt}}$
$\dfrac{{dV}}{{dt}} = 4\pi {r^2} \times \dfrac{1}{{8\pi r}}\dfrac{{dS}}{{dt}}$
After simplifying
$\dfrac{{dV}}{{dt}} = \dfrac{r}{2}\dfrac{{dS}}{{dt}}$
$\dfrac{{dV}}{{dt}} = \dfrac{1}{2}r\dfrac{{dS}}{{dt}}$

Hence, the correct option is B.

Note: Students should use the correct formula to avoid any calculation errors. They should solve questions step by step for the correct answer. While differentiating they should apply rules of differentiation. For an easy solution first, they differentiate the surface area and then the volume of the sphere so that they get the required answer.