Reltated Rates: how fast is the water level rising?

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I have a big test this week, so I've been going over related rates problems. I've been getting all of them except for this one which, for some reason, is totally stumping me. Here goes:

A spherical tank has a radius of 10 ft. It is being filled with water at a rate of 200 gal/min. How fast is the water level rising when the max depth of water in the tank is 5 ft?

. . .V = (1/3)(pi)(y<sup>2</sup>)(3a - y)

...where "y" is the depth in feet.

There is a picture in our book of the sphere. It shows that "y" is the depth, and it has a line for the radius. The radius connects what looks like the middle of y (but I don't think it is the exact middle) to make a right angle. The hypotenuse is "a".

If anyone could tell me where to start, I would greatly appreciate it. Thank you!
 
Use the volume of a sphere, not a cone.

\(\displaystyle \L\\\frac{4}{3}{\pi}r^{3}\)
 
aswimmer113 said:
A spherical tank has a radius of 10 ft. It is being filled with water at a rate of 200 gal/min. How fast is the water level rising when the max depth of water in the tank is 5 ft?
Click to expand...
They don't ALL require calculus.

Try Rate / Surface Area

The Pythagorean Theorem provides the radius: \(\displaystyle \sqrt{10^{2}-5^{2}} = 5*\sqrt{3}\)

Rate = 200 gal/min
Surface Area = \(\displaystyle \pi*(5*\sqrt{3})^{2}\)

It's not a general solution.
 
Tkh's method is simplistic and the way to go, but if you must use

calculus....I was thinking about the volume of a spherical cap. Let the

water at height h be the cap.

Edit: Due to wjm11's perspicaciousness, we should convert gallons to

cubic units. I will use cubic feet. There are approx. 7.48 gallon in a cubic

foot.That was tricky of them to throw that in there like that. Good catch,

wjm11.

\(\displaystyle \frac{200}{(\frac{187}{25})}=\frac{5000}{187}ft^{3}\min.\)

\(\displaystyle \L\\\text{Volume of cap=}\frac{{\pi}h^{2}(3r-h)}{3}\)

Differentiate:

\(\displaystyle \L\\\frac{dV}{dh}=(2rh-h^{2}){\pi}\frac{dh}{dt}\)

enter in given values and solve for dh/dt:

\(\displaystyle \L\\(2(10)(5)-25){\pi}\frac{dh}{dt}=\frac{5000}{187}\)

\(\displaystyle \L\\\frac{dh}{dt}=\frac{\frac{5000}{187}}{75{\pi}}=\frac{200}{561{\pi}}\approx{.113}\ \text{ft/min}\)

Or about \(\displaystyle 1\frac{3}{8}\) in/min.
 
A spherical tank has a radius of 10 ft. It is being filled with water at a rate of 200 gal/min. How fast is the water level rising when the max depth of water in the tank is 5 ft?
Click to expand...

Keep an eye on your units. You want your answer to be in in/min or ft/min. The "200" is in gal/min.
 
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