AP Calculus AB · Differential equations
A tank contains 100 gallons of water at time t=0 minutes
Problem
A tank contains \(100\) gallons of water at time \(t=0\) minutes. Water flows into the tank at the constant rate of \(30\) gallons per minute and flows out at the rate of \(2t\) gallons per minute for \(0\le t\le 20\). The tank has a capacity of \(400\) gallons. Let \(V(t)\) be the number of gallons in the tank at time \(t\). (a) Write a formula for \(V'(t)\) and a definite-integral formula for \(V(t)\). (b) Find every time in \([0,20]\) at which \(V\) has a critical point, and determine the corresponding amount of water in the tank. (c) Find the maximum amount of water in the tank on \(0\le t\le 20\). Does the tank overflow? Give a reason. (d) How many gallons of water flow out of the tank from \(t=0\) to the time at which \(V\) attains its maximum?
Hint
Net rate is in minus out; a critical point of \(V\) is a zero of that net rate; overflow is a comparison with capacity.
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