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Question

At t = 0 minutes, a tank contains 48 gallons of water. For 0 ≤ t ≤ 4 minutes, water flows into the tank at a rate of 12T sin (pi/4xT) gallons per minute, and water leaks out of the tank at a rate of 6T^2/T+2 gallons per minute. How many gallons of water are in the tank at the time t = 4 minutes? 

do I submit 4 in for t I don't think I do that but i don't know what to do for this

Answer 1

You are correct that you can't just plug 4 in for t into the two rate functions. All that will tell you are the flow rates in and out at t = 4, not the volume. Instead, we use t = 4 as the upper bound of integration. We need need to integrate the net rate (flow rate in - flow rate out) over the time interval to get the total change in volume at that time. Because the tanks starts with 48 gallons in it, V_t=4 = 48 + ∫₀⁴ (rate in - rate out)dt .