Integrals Featuring John Adams

Cheesecake and John Adams... How does one combine these two great tastes?

A historically correct hypothetical situation in which both objects are located in the royal courts of France is the way to go.

 The man... the legend... 

John Adams, as we all know, went to France as a diplomat during the Revolutionary War. As he was there, he experienced French court life which he disliked due to the slowness of their political system. 

One day, as Adams attended a court dinner in which a 18th century equivalent of cheesecake was served. Enthralled with the exquisite food, Adams began eating it at a rate expressed by:

Where c(t) expresses the rate at which cheesecake is being consumed in bites and 't' is in time in minutes. The equation is also bounded by:

Also, at t = 0 minutes, Adams had not taken any bites of cheesecake mostly because it is hard to eat cheesecake before its served.

(a) Write the the indefinite integral for the rate at which cheesecake is being consumed. Explain what this equation represents. (Hint: make sure you find a way to solve for C using the given information)

(b) Find out the rate at which John Adams is eating cheesecake at t = 8 minutes. Indicate units.

(c) Find the amount of cheesecake that John Adams has consumed between t = 3 and t = 14 minutes. Indicate units.

(d) Assuming that Adams finishes eating his cheesecake at t = 20 minutes and each bite that he took was 5 grams worth of cheesecake, determine the mass of the cheesecake. 

Solutions:

The rate that is given for the consumption of cheesecake is that way because people generally eat at a fast rate and then slow down as they get closer to being full. 

Now, Adams likes to savor his food, which is why he took 20 minutes to eat his cheesecake. 

(a) 

By using the "reverse power rule" you get:

Now, after you take the integral of the rate of something you get the amount of something. Since you just took the integral of the rate of cheesecake consumption in "bites", then the integral represents how much cheesecake has been eaten. In order to solve for "C", you must refer to the information given on how much cheesecake has been consumed. In this case, Adams has eaten 0 bites of cheesecake at t = 0 minutes.

Therefore, you can set the integral equal to 0 and set t = 0. From this, you can see that C = 0.

(b) 6 bites / minute

This is asking for the rate of consumption of cheesecake by Adams. That equation is already given, therefore plug and chug fellows... plug and chug...

The units are bites / min because that is the units that are given for the rate at which John Adams eats cheesecake.

(c) 63.25 bites

If you figured out part (a), then more than half of this problem has already been solved. In order to get the amount eaten, you must take a definite integral with the bounds given:

The unit is bites because if you take the integral of a rate then you get the amount.

 

That's averaging about 5.75 bites a minute. Or almost one bite every ten seconds. Adams like his cheesecake.

(d) 500 grams

 

This requires another definite integral, but this time from t = 0 to t = 20 because you want to know how many bites of cheesecake Adams has taken over the whole period of time he has been eating:

Then you have to take the number of bites and multiply by 5 grams in order to get the total mass of the cheesecake.

Questions? Corrections? Email me at Jonathan352@gmail.com or comment!