How to find the rate at which water is being pumped into the tank in cubic centimeters per minute? details below:
Water is leaking out of an inverted conical tank at a rate of #6700# cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height #7# meters and the diameter at the top is #4.5# meters. If the water level is rising at a rate of #20# centimeters per minute when the height of the water is
#3# meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
Note: Let "#R# " be the unknown rate at which water is being pumped in. Then you know that if #V# is volume of water, #dV/dt =R−6700# . Use geometry (similar triangles?) to find the relationship between the height of the water and the volume of the water at any given time. Recall that the volume of a cone with base radius #r# and height #h# is given by
#1/3 pi r^2h# .
Water is leaking out of an inverted conical tank at a rate of
Note: Let "
1 Answer
Explanation:
The notes provided definitely set you on the path towards solving the problem. As noted, the net
This allows us to take the derivative of
This is where the similar triangles concept comes into play. If the cone (at its widest point at the top) has a diameter
Furthermore, we know that the ratio of the radius
We can now substitute this into the formula for the volume:
Now we have a volume formula we can work with. We can take derivatives with respect to time
There are 3 quantities in this expression we already know:
We can substitute all of these and solve for