Water is poured into a hemispherical bowl of radius 10 cm at the rate of \(15\pi\) cm3/s. When the depth of water is h cm, the volume of water in the bowl is \(V = \displaystyle\frac{\pi h^{2}(30 - h)}{3}\) cm3. The rate at which the water level is rising when the depth of water is 5 cm is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{5}\) cm/s
Writing the volume as a polynomial in h:
\( \displaystyle V = \frac{\pi}{3}\left(30h^{2} - h^{3}\right) \)
Differentiating with respect to time t by the chain rule:
\( \displaystyle \frac{dV}{dt} = \frac{\pi}{3}\left(60h - 3h^{2}\right)\frac{dh}{dt} \)
\( \displaystyle = \pi\left(20h - h^{2}\right)\frac{dh}{dt} \)
When h = 5:
\( \pi(100 - 25) = 75\pi \)
With \(\frac{dV}{dt} = 15\pi\):
\( \displaystyle \frac{dh}{dt} = \frac{15\pi}{75\pi} = \frac{1}{5} \) cm/s