First slide

Diffrentiation

Question

The radius of an air bubble is increasing at the rate of $\frac{1}{2}$ cms^-1. Determine the rate of increase in it's volume when the radius is 1 cm.

Moderate

A

6.29 cm²s²
B

6.28 cm⁴s^-3
C

6.28 cm³s^-1
D

6.28 cm³s^-4

Solution

Volume of the spherical bubble $v = \frac{4}{3} {πR}^{3}$
differentiating both sides w.r.t. time
$\frac{dV}{dt} = \frac{d}{dt} (\frac{4}{3} {πR}^{3})$
$\Rightarrow \frac{dV}{dt} = (\frac{4}{3} π) (3 R^{2} \frac{dR}{dt})$
$\Rightarrow \frac{dV}{dt} = 4 {πR}^{2} \frac{dR}{dt}$
at $R = 1$ cm and when $\frac{dR}{dt} = \frac{1}{2} {cms}^{- 1}$ , we have
$\begin{array}{rcl} \frac{dV}{dt} & = & 4 π \times {(1)}^{2} \times \frac{1}{2} \\ = & 2 π {cm}^{3} s^{- 1} \\ = & 6.28 {cm}^{3} s^{- 1} \end{array}$

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