Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time water is being pumped into the tank at a constant rate If the tank has a height of 6m and the diameter at the top is 4 m and if the water level is rising at a rate of 20 cm/min when the height of the water is 2m, how do you find the rate at which the water is being pumped into the tank?

Answer 1
Let #V# be the volume of water in the tank, in #cm^3#; let #h# be the depth/height of the water, in cm; and let #r# be the radius of the surface of the water (on top), in cm. Since the tank is an inverted cone, so is the mass of water. Since the tank has a height of 6 m and a radius at the top of 2 m, similar triangles implies that #\frac{h}{r}=\frac{6}{2}=3# so that #h=3r#.
The volume of the inverted cone of water is then #V=\frac{1}{3}\pi r^{2}h=\pi r^{3}#.
Now differentiate both sides with respect to time #t# (in minutes) to get #\frac{dV}{dt}=3\pi r^{2}\cdot \frac{dr}{dt}# (the Chain Rule is used in this step).
If #V_{i}# is the volume of water that has been pumped in, then #\frac{dV}{dt}=\frac{dV_{i}}{dt}-10000=3\pi\cdot (\frac{200}{3})^{2}\cdot 20# (when the height/depth of water is 2 meters, the radius of the water is #\frac{200}{3}# cm).
Therefore #\frac{dV_{i}}{dt}=\frac{800000\pi}{3}+10000\approx 847758\ \frac{\mbox{cm}^3}{min}#.
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Answer 2

The rate at which the water is being pumped into the tank is ( 400 \pi , \text{cm}^3/\text{min} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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