Question

A cylindrical water tank is being filled at a rate of LaTeX: 10\:m^3\:10 m 3 per minute and is leaking at a rate of LaTeX: 3\:m^33 m 3 per minute. If the water level is changing at a rate of LaTeX: \frac{4}{\pi}4 π meters/minute when the height is at 6 meters. What is the radius of the tank?

Answer 1

Answer:

1.323 m

Explanation:

Rate of filling = $10\ \text{m}^3/\text{min}$

Rate of leakage = $3\ \text{m}^3/\text{min}$

Net change in volume of water = $10-3=7\ \text{m}^3/\text{min}$

$\dfrac{dh}{dt}$ = Rate of change of height = $\dfrac{4}{\pi}\ \text{m/min}$

Volume of cylinder is given by

$V=\pi r^2h$

Differentiating with respect to time we get

$\dfrac{dV}{dt}=\pi r^2\dfrac{dh}{dt}\\\Rightarrow 7=\pi r^2\times \dfrac{4}{\pi}\\\Rightarrow r=\sqrt{\dfrac{7}{4}}\\\Rightarrow r=1.323\ \text{m}$

The radius of the tank is 1.323 m.