2005 FR 5 ( i edited my mistakes)

The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by
A pumping station adds sand to the beach at a rate modeled by the function S, given by

Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for . At time t=0, the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.

fnInt(2+5sin(4π t/25),t,0,6)= 31.816 cubic yards

(b) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.

Y(t) = fnInt(s(t))- fnInt(R(t)) + 2500

Y(t) = fnInt((15t/(1+3t)) - (2+5sin(4π t/25)) + 2500

(c) Find the rate at which the total amount of sand on the beach is changing at time t=4.

(Y(t))=nDeriv (2500+fnInt (S(t)-R(t), t, 0, x) )=S(t)-R(t).

S(4)-R(4)=-1.909 cubic yards/hours.

(d) For , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.

S(t)-R(t) = 0 to find critical pts.

critical pt at t=5.118,and 0 and 6

y(5.118)= 2492.369 cubic yards

t=5.118

minimum value at t=5.118 is 2 492.369 cubic yards

Thank you!!!

for correcting my mistakes!!!