# Thread: Tide removing sand from a beach

1. ## Tide removing sand from a beach

The tide removes the sand from a beach at a rate modeled by the function R, given by$\displaystyle R(t) = 2+5sin\frac{4*pi*t}{25}$. A pumping station adds sand to the beach at a rate modeled by the function S, given by $\displaystyle S(t)=(15t)/(1+3t)$.

Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for $\displaystyle 0=<t=<6$. At time $\displaystyle t=0$, the beach contains 2500 cubic yards of sand.

How much sand will the tide remove from the beach during this 6 hour period. Indicate units of measure. Write an expression for the total number of cubic yards of sand on the beach at time t.

Any help is appreciated, thanks.

2. Originally Posted by lancelot854
The tide removes the sand from a beach at a rate modeled by the function R, given by$\displaystyle R(t) = 2+5sin\frac{4*pi*t}{25}$. A pumping station adds sand to the beach at a rate modeled by the function S, given by $\displaystyle S(t)=(15t)/(1+3t)$.

Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for $\displaystyle 0=<t=<6$. At time $\displaystyle t=0$, the beach contains 2500 cubic yards of sand.

How much sand will the tide remove from the beach during this 6 hour period. Indicate units of measure. Write an expression for the total number of cubic yards of sand on the beach at time t.
sand removed from the beach in 6 hrs ... $\displaystyle \displaystyle \int_0^6 R(t) \, dt$

total sand on the beach at time t ... $\displaystyle \displaystyle 2500 + \int_0^t S(x) - R(x) \, dx$

both accumulation functions will have units in cubic yards of sand.

3. I knew this looked familiar. It's from the AP Calculus AB 2005 Free Response questions. I'm feeling a little too lazy to solve this right now, but if you Google "Lou Talman" you'll find the answer to this question and a lot others (so I can be justifiably called efficient instead of lazy...yeah, that's it).

He only gives solutions but the questions themselves can all be found on the AP site.