Advanced Placement Cheesecake: Alligator Edition

Now that many of us have gone through the trauma and ordeal of taking the AP Calculus test, and it's been long enough for me to not jeopardize my AP scores, it is time to begin working the free response questions.

If you would like to find the original 2010 Calculus AB problems they are here:

2010 AP Calculus AB Free Response

Our first question will involve a certain alligator owned by John Quincy Adams:

 Sadly, it wasn't an albino alligator like this one

For convenience sake, the alligator will be from now on referred to as "Tim".

John Quincy Adams's alligator had a voracious appetite for cheesecake. There is no cheesecake in his food bowl before 12 P.M (t = 0). At precisely 12 P.M. John Quincy Adams begins to fill Tim's food bowl with cheesecake at a rate of:

Where t is in hours and f(t) is in cubic inches per hour.

Tim then begins eating the cheesecake at this rate:

 (a) How much cheesecake is in Tim's food bowl at 6 P.M.?

What the AP test writers are testing you over is whether you realize you need to take the integral of a rate to get the amount. Since from zero to six hours there is no cheesecake eaten and only cheesecake being added at a rate of f(t), all you have to do is take the definite integral of f(t) from zero to six:

Your units are feet cubed because the integral of a rate, inches cubed per hour, gets you the amount, which is inches cubed.

(b) Find the rate of change of cheesecake in Tim's food bowl at 8 A.M.

This question is testing your ability to realize that the question already gave you the rate and that you have to account for the addition of cheesecake and the removal of cheesecake. You should get something like this:

 

Units are inches cubed per hours because that is the rate given in the problem. The negative sign indicates that the amount of cheesecake is decreasing at that time. The negative sign is not necessary if you indicated that cheesecake was being removed or was decreasing at that rate.

(c) Let h(t) represent the total amount of cheesecake in cubic inches that Tim has eaten at t hours after noon. Express h as a piecewise function with domain zero to nine.

This is yet another problem testing your ability to recognize that you need to take the integral with a twist: this time you have to integrate a piecewise function. Don't fear though, all you have to do is integrate each separate part and then write it as three separate functions which should look something like this:

(d) How much cheesecake is in Tim's food bowl at 9 P.M.?

Yet again, more integrals, but this time you had to realize that you have to include the integrals of the rate of addition of cheesecake and subtract from that the integral of the consumption of cheesecake. This should look like:

The expression can be written in several different ways, but your answer should be the same no matter what.

Tim likes his cheesecake...

I will follow with the rest of the AP problems... soon.

If you have any questions or concerns about this problem, feel free to contact me at Jonathan352@gmail.com