Question:
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
(a) $1 \mathrm{~m} / \mathrm{hr}$
(b) $0.1 \mathrm{~m} / \mathrm{hr}$
(c) $1.1 \mathrm{~m} / \mathrm{hr}$
(d) $0.5 \mathrm{~m} / \mathrm{hr}$
Solution:
(a) 1 m/hr
Let $r, h$ and $V$ be the radius, height and volume of the cylinder at any time $t .$ Then,
$V=\pi r^{2} h$
$\Rightarrow \frac{d V}{d t}=\pi r^{2} \frac{d h}{d t}$
$\Rightarrow 314=3.14 \times(10)^{2} \frac{d h}{d t}$
$\Rightarrow \frac{d h}{d t}=\frac{314}{314}$
$\Rightarrow \frac{d h}{d t}=1 \mathrm{~m} / \mathrm{hr}$