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

1.323 m

Step-by-step 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}`

`(dh)/(dt)` = Rate of change of height =
`(4)/(\pi)\ \text{m/min}`

Volume of cylinder is given by

`V=\pi r^2h`

Differentiating with respect to time we get

`(dV)/(dt)=\pi r^2(dh)/(dt)\\\Rightarrow 7=\pi r^2* (4)/(\pi)\\\Rightarrow r=\sqrt{(7)/(4)}\\\Rightarrow r=1.323\ \text{m}`

The radius of the tank is 1.323 m.