Cylindrical tank rate of change Water is pouring into a cylinder with a radius of 5m and height of 20m at a rate of 3 cubic metres a minute. Find the rate of change of height when the tank is half full. Now the Volume V = pir^2h and I can determine the rate of change in Volume is dV/dt=pir^2dh/dt and the rate of change of height is dh/dt=1/pir^2 yimes dV/dt Using that formula I can determine that the water is rising at a rate of 3/25pi m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?

Ivan Buckley 2022-09-23 Answered
Cylindrical tank rate of change
Water is pouring into a cylinder with a radius of 5m and height of 20m at a rate of 3 cubic metres a minute. Find the rate of change of height when the tank is half full.
Now the Volume V = π r 2 h and I can determine the rate of change in Volume is d V / d t = π r 2 d h / d t and the rate of change of height is d h / d t = 1 / π r 2 × d V / d t
Using that formula I can determine that the water is rising at a rate of 3 / 25 π m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?
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Answers (1)

niveaus7s
Answered 2022-09-24 Author has 8 answers
You have correctly identified that water level is rising at the rate of d h d t = 3 25 π m/min.
Observe that 3 25 π does not depend on the implicit variable t (the time in minutes), therefore the rate of change of height is the same when the tank is half full or full. You've already found it.
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