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NatyBaby's question at Yahoo! Answers regarding finding the rate water is pumped into a tank

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MarkFL

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Feb 24, 2012
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Here is the question:

Related Rates Question: Water is leaking out of an inverted conical tank at a rate of 8,500 cm^3/min...?Help!?


Water is leaking out of an inverted conical tank at a rate of 8,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (Round your answer to the nearest integer.)

Please and thank you!
I have posted a link there to this topic so the OP can see my work.
 
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MarkFL

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Feb 24, 2012
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Hello NatyBaby,

Let all linear measures be in centimeters and time be measured in minutes.

The statement:

"Water is leaking out of an inverted conical tank at a rate of 8,500 cm3/min at the same time that water is being pumped into the tank at a constant rate."

\(\displaystyle \frac{dV}{dt}=R-8500\)

Where $V$ is the volume of the water in the tank at time $t$ and $R$ is the rate at which water is being pumped into the tank. $R$ is the quantity we are asked to find.

Using the volume of a cone, we know the volume of the water in the tank may be given by:

\(\displaystyle V=\frac{1}{3}\pi r^2h\)

where $r$ is the radius of the surface of the water, and $h$ is the depth of the water. We know that at any given time or volume of water, the ratio of the radius of the water at the surface to its depth will remain constant, and in fact will be in the same proportions as the tank itself.

Because we are given information regarding the time rate of change of the depth and the depth itself, we need to place the radius with a function of the depth. Hence, we may use:

\(\displaystyle \frac{r}{h}=\frac{2}{6}=\frac{1}{3}\implies r=\frac{h}{3}\)

And so the volume as a function of $h$ is:

\(\displaystyle V=\frac{1}{3}\pi \left(\frac{h}{3} \right)^2h=\frac{\pi}{27}h^3\)

Now, differentiating with respect to $t$, we obtain:

\(\displaystyle \frac{dV}{dt}=\frac{\pi}{9}h^2\frac{dh}{dt}\)

Equating the two expressions for \(\displaystyle \frac{dV}{dt}\), we find:

\(\displaystyle \frac{\pi}{9}h^2\frac{dh}{dt}=R-8500\)

Solving for $R$, we get:

\(\displaystyle R=\frac{\pi}{9}h^2\frac{dh}{dt}+8500\)

Now, using the given data (making sure all of our units match):

\(\displaystyle h=200\text{ cm},\,\frac{dh}{dt}=20\,\frac{\text{cm}}{\text{min}}\)

We have:

\(\displaystyle R=\frac{\pi}{9}\left(200\text{ cm} \right)^2\left(20\,\frac{\text{cm}}{\text{min}} \right)+8500\,\frac{\text{cm}^3}{\text{min}}\)

\(\displaystyle R=\frac{500}{9}\left(1600\pi+153 \right)\,\frac{\text{cm}^3}{\text{min}}\)

Rounded to the nearest integer this is:

\(\displaystyle R\approx287753\,\frac{\text{cm}^3}{\text{min}}\)