A large cistern is being filled with water. The inflow rate, measured in liters per minute, is modeled for the first 30 minutes by the continuous function
r(t)=4+3sin(6πt),0≤t≤30,
where t is the time in minutes after filling begins.
Assume the cistern is empty at t=0.
(a) Write an expression, in terms of a definite integral, for the volume of water V(t) (in liters) that has entered the cistern during the interval 0≤t≤t. Do not evaluate the integral.
(b) Evaluate the integral from part (a) to obtain an explicit formula for V(t).
(c) Determine the time T, with 0<T<30, at which exactly 500 liters of water have entered the cistern. Show all steps leading to your result, and give the answer correct to the nearest tenth of a minute.
(d) Interpret the meaning of V′(T) in the context of this problem, and compute its value at the time found in part (c). Include appropriate units.