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If the volume of a hemisphere is increasing at a steady r...

If the volume of a hemisphere is increasing at a steady rate of 18π m $^{3}$ s $^{-1}$ , at what rate is its radius changing when its is 6m?

A.

2.50m/s

B.

2.00 m/s

C.

0.25 m/s

D.

0.20 m/s

View answer & explanation

Correct answer is C

$V = \frac{2}{3} \pi r^{3}$

Given: $\frac{\mathrm d V}{\mathrm d t} = 18\pi m^{3} s^{-1}$

$\frac{\mathrm d V}{\mathrm d t} = \frac{\mathrm d V}{\mathrm d r} \times \frac{\mathrm d r}{\mathrm d t}$

$\frac{\mathrm d V}{\mathrm d r} = 2\pi r^{2}$

$18\pi = 2\pi r^{2} \times \frac{\mathrm d r}{\mathrm d t}$

$\frac{\mathrm d r}{\mathrm d t} = \frac{18\pi}{2\pi r^{2}} = \frac{9}{r^{2}}$

The rate of change of the radius when r = 6m,

$\frac{\mathrm d r}{\mathrm d t} = \frac{9}{6^{2}} = \frac{1}{4}$

= $0.25 ms^{-1}$

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