A city work fills the cylindrical tank of a new water tower. The tank has a radius of 7 feet. At time t minutes, the height of the water in the tank is t4 feet and the volume of water in the water tower is V=πr2h cubic feet. Find the instantaneous rate of change of the volume of water in the tank at t=10 minutes.
To find the instantaneous rate of change of the volume of water in the tank at t=10 minutes, we need to find the derivative of the volume function with respect to time and evaluate it at t=10 minutes.
Given that the radius of the tank is 7 feet and the height of the water at time t is t^4 feet, we can find the volume V of the water at time t using the formula V = πr^2h.
Substituting the given values, we have V = π(7^2)t^4 = 49πt^4.
To find the derivative of V with respect to t, we can apply the power rule for differentiation. The power rule states that the derivative of t^n with respect to t is n*t^(n-1).
Using the power rule, the derivative of V with respect to t is dV/dt = 4*49πt^3 = 196πt^3.
To find the instantaneous rate of change of the volume at t=10 minutes, we substitute t=10 into the derivative:
dV/dt = 196π(10)^3 = 196π*1000 = 196000π.
Therefore, the instantaneous rate of change of the volume of water in the tank at t=10 minutes is 196000π cubic feet per minute.