[SOLVED] Find rate of increase of water level?

**fardeen_gen** · June 29th 2008, 07:44 AM

Water runs into a conical tank at the rate of 9 ft^3/ min. The tank stands point down and has height of 10 ft and a base radius of 5 ft. How fast in the water level rising when water is 6 feet deep?

What is the answer? Please explain the steps also.

**galactus** · June 29th 2008, 07:49 AM

This is an old cliche related rates problem. Do a site search and many will pop up.

**Isomorphism** · June 29th 2008, 07:53 AM

Originally Posted by fardeen_gen

Water runs into a conical tank at the rate of 9 ft^3/ min. The tank stands point down and has height of 10 ft and a base radius of 5 ft. How fast in the water level rising when water is 6 feet deep?

What is the answer? Please explain the steps also.

$V = \frac13 \pi r^2 h$

Now when the water is at a level of h, let the surface water radius be r. Now if you draw a figure, you will see that if the half angle of the cone is $\theta$ , then $\tan \theta = \frac{5}{10} = \frac12 = \frac{r}{h}$

Thus $V = \frac13 \pi \left(\frac{h}2\right)^2 h = \frac{\pi h^3}{12} \Rightarrow \frac{dV}{dt} = \frac{3\pi h^2}{12}\bigg{|}_{h=6}\frac{dh}{dt}$

$9 = \frac{\pi 6^2}{4}\frac{dh}{dt} \Rightarrow \frac{dh}{dt} = \frac1{\pi} \text{ ft/min}$

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