Water draining from a tank

Question

The tank holds 3500 gallons of water. The volume of the tank is given by v=3500(1-(t/50))^2, t between 0 and 50

A) At what time is the water flowing out the fastest?

B) The slowest?

Thank you!

Answer 1

You need to take the derivative of Volume, dV/dt and equate it to 0 and where it blows up in order to find critical points, then check for sign change in the derivative or 2nd derivative to identify min or max. You also need to check boundary points for the interval. This problem does not have any quirks and can be solved by inspection.

dV/dt = 3500(2)(1-t/50)(-1/50) = -140(1-t/50) Only critical point is t = 50 where the tank is emply. (obvious min)

The max must be the other boundary, or t=0 You can calculate the rate by plugging into dV/dt equation.