Rates of Change — Question 3

Question 3

Problem:

Water is being poured into a cylindrical tank at a rate of $500 \, \text{cm}^3/\text{s}$. The tank has a radius of $10 \, \text{cm}$. How fast is the water level rising when the water is $20 \, \text{cm}$ deep?

Question 3 - Solution

The volume of a cylinder is: $$V = \pi r^2 h$$

Given: $\frac{dV}{dt} = 500$ , $r = 10$ , $h = 20$

We differentiate with respect to time: $$\frac{dV}{dt} = \pi r^2 \frac{dh}{dt} \Rightarrow 500 = \pi (10)^2 \frac{dh}{dt} \Rightarrow 500 = 100\pi \frac{dh}{dt} \Rightarrow \frac{dh}{dt} = \frac{500}{100\pi} = \frac{5}{\pi}$$

Answer: $$\boxed{\frac{dh}{dt} = \frac{5}{\pi} \text{ cm/sec}}$$