The Conundrum of Rotating Cakes

AP Calculus AB 2010 Free Response #4 is quite the conundrum, but cakes, once again, will analogically save the day.

The problem has to do with this:

But cake makes it better so:

Let "R" (or the cake) be the region in the first quadrant bounded by the graph of y = 2√(x), the horizontal line y = 6, and the y axis.

(a) Find the area of "R", or, in this case, the area of the cake.

The best way to think about this is to imagine the cake being a whole rectangle:

Now you can recognize that the area of the cake is the area of the whole cake minus the area that the y = 2√(x) cuts out. The area of the cake is 54 units through multiplication, but ,in order to do this the Calculus way, you'll need to subtract the area under the curve (or integral) of the y = 2√(x) from the area under the curve (or integral) of y = 6 from x = 0 to x = 9:

(b) Write, but do not evaluate, an integral expression that gives the volume of the cake generated when "R" is rotated about the horizontal line y = 7.

This is where it gets much more tricky. Remember that there are three ways to integrate solids generated by rotation: disk, washer, and the shell method. In this case, we will have to use the washer method because, like a washer, there will be a hole in the center of the solid. You can tell this because you're rotating it at y = 7 which is greater than the y = 6 that the graph extends to. Therefore, you know that there will be a hole in between y = 6 and y = 7.

The rotation would look something like this:

Rotate along y = 7, get cake.

Now, you must determine how to use the washer method. Remember that the format for it is like this:

It is possible to reason through and get this equation if you don't memorize it. Remember that the 'pi' is there because you're rotating something in a circle, the integral is the domain (or range in case of dy) of the thing you are rotating, the outer and inner radii are squared which is similar to squaring the radius in order to get the area of a circle, and you subtract the outer radius from the inner radius because of the hole that is left by the inner radius. 

Now all we have to do is determine our "outer" radius which is:

The reason why this is the outer is because it gives the equation for all of the area under y = 7, but not the 'hole' on the inside. The hole on the inside, when rotated, is just a cylinder with a radius of r = 1, so all the equation for the inside radius is simply:

Now all you have to do is plug into the equation, and you're done:

(c) Region R is the base of a cake. For each y, where 0 < y < 6, the cross section of the cake taken
perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write,
but do not evaluate, an integral expression that gives the volume of the solid.

For this, we will refer back to this picture:

Now all you have to do is imagine that there are a large number of rectangles stacked on top of each other going from left to right on the cake portion of the graph. In order to calculate what the estimate of a volume these rectangles is, you must first find them in terms of 'dy' because they are perpendicular to the y axis. To do this, solve the original equation for 'x':

And since the problem tells you that the height of these rectangles is three times greater than the base, then:

Now, with the height and the base, you can find the area of ONE of the rectangles:

After getting the equation for ONE of the rectangles, the full rectangular estimate of the volume can be found by taking the integral of this equation:

The integral is taken from 0 to 6 because the base is perpendicular to the y axis and that base goes from 0 to 6. The integral is needed because it adds up all of the infinitely thin rectangles within the volume of the cake in order to get an estimate of the actual volume.

This problem was extremely 'thick'. If you need any clarification on the concepts needed to solve this problem or if you have any corrections or suggestions, do not hesitate to contact me at jonathan352@gmail.com.

A Certain Famous Virginian...

No, not George Washington but THIS man:

"Here a Lee, there a Lee... Social-Lee, Political-Lee, Financial-Lee"

The above is clearly a dramatization (a la 1776 the Musical) THIS, is what Richard Henry Lee most likely looked like:

Orange attire not included 

  

Lee was instrumental in the adoption of the Declaration of Independence because he was from Virginia, one of the richest colonies at the time, which gave the opportunity to put forth the motion for the Declaration and gain support for it.

In order to gain Virginian support, he gave a speech to the House of Burgesses:

Before Lee arrived, there were already 700 people in the crowd. During the day, people joined the crowd according to this graph for a function r (t):

Eight hour speeches were the norm back in the day...

 

People also leave the crowd at a rate of 800 people per hour.

(a) How many people join the crowd between t = 0 and t = 3?

In order to understand how to approach this problem, you must realize that the graph they have given you relates the rate of people entering the crowd to time. The question asks for the amount of people that join the crowd, so therefore you must take the definite integral (or the area under the curve) from t = 0 to t = 3 to find out the amount of people that has joined the crowd from the rate graph they give you:

Since the integral is the area under the curve, using the area of a trapezoid formula would have net you the correct answer. Also note that the original 700 people that were already there do not contribute to the answer as the question asks for how many people ARRIVE.

(b) Is the number of people in the crowd increasing or decreasing between t = 2 and t  = 3?

Due to Richard Henry Lee's charisma (and the graph), the size of the crowd is still increasing between the second and third hour. According to the graph, more than 800 people are joining the crowd between t = 2 and t = 3. Since this is greater than the 800 people per hour who are leaving, the size of the crowd must be getting larger.

(c) At what time "t" is the size of the crowd the largest? How many people are in the line at that time?

The point at which the crowd is the largest is when the size of the crowd goes from growing to shrinking. Since 800 people are leaving each hour, you must find the point on the graph at which the people joining the crowd is less than 800 which is at t = 3.

The second part of the question must be answered with a definite integral because it is asking you to find the amount people in the line from the graph of the rate at which people are joining the crowd. Now you must take into account the 700 people that were originally there and the 800 people that leave every hour. This means that you must subtract the definite integral from t = 0 to t = 3 of 800 from the definite integral of the graph and add 700:

(d) Write, but do not solve, an equation involving the integral expression of r whose solution gives the earliest time "t" for which there is no longer a crowd.

This follows the same principal as the previous problem. Add the integral of the graph, subtract the integral of the 800, and add 700. Remember to set your equation equal to 0 as that is what the question is asking:

His speech probably went something like this:

Questions? Concerns? Contact me at Jonathan352@gmail.com

AP Cheesecake: The Many Duels of Andrew Jackson

We now continue on our journey of working the 2010 AP Calculus questions with question number two.

2010 AP Calculus AB Free Response Questions

Our story now moves from John Quincy Adams's albino alligator Tim, to the many duels of Andrew Jackson.

As we all know, Andrew Jackson was a badass. Among his many achievements are:

- First president to be attacked while president
- First president to have an assassination attempt
- Survived 13 duels, injured many times, killed a man
- Stopped Civil War between Michigan and Ohio
- Destroyed the 2nd Bank of the United States
- Enlistment of pirates (technically privateers) during Battle of New Orleans

Dramatization

- Ironically ended up on this even though he directly caused the Panic of 1837:

1905 George Washington $20 bill was so much better

One fact that has been lost in history is that Jackson enjoyed cheesecake, especially before and after duels. In fact, he used to go on cheesecake binges before duels.

In one of Jackson's recently uncovered journals, he recorded the amount of cheesecake he consumed:

(a) Approximate the rate at which Andrew Jackson is consuming cheesecake, in hundreds of grams per hour, at t = 6.

Since, for some reason, Jackson decided to only record the amount of cheesecake he ate at odd intervals (probably because of the cheesecake induced stupor), we will have to "approximate", as the problem says, the rate at which Jackson is eating cheesecake.

The two tricks to realize here are that it's asking you for the rate, and it asks you to approximate. We have the chart of the amount, so in order to get the rate we will need the derivative, but since there's no way to take the derivative of this we will have to approximate it by using the slope between t = 5 and t = 7:

 Units are in hundreds of grams of cheesecake per hour because that is what the slope would be of a line created by a plot of hours vs hundreds of grams of cheesecake.

(b) Use a trapezoidal sum with four sub-intervals given by the table in order to approximate:

Using correct units, explain the meaning of this in terms of number of entrees.

This question tests your knowledge of how a trapezoidal Riemann Sum works and you knowledge of understanding how an integral works in this context.

 Remember that the formula for a trapezoid is:

In order to take the Riemann Sum, we must calculate the areas of the four rectangles the table in the problem forms:

Which adds up to around 10.688 hundred grams of cheesecake per hour. The 1/8 serves to indicate that this is an average of the amount of cheesecake consumed in hundreds of grams per hour from t = 0 to t = 8.


(c) At 8 P.M., Jackson began the painful process of throwing up cheesecake at a rate of:

If this occurs from t = 8 to t = 12, how much cheesecake is still inside Jackson at t = 12?

In order to solve this problem, you must realize that what you need to find is an amount, and what they give you is one amount and one rate. The amount of cheesecake Jackson has consumed at t = 8 is given in the chart, and the rate of expulsion of cheesecake must be subtracted from that. In order to get an amount of cheesecake, we must take the integral of the rate from t = 8 to t = 12 which is to be subtracted from the amount of cheesecake at t = 8:

(d) According to the model from part (c), at what time is Jackson upchucking cheesecake at the greatest rate?

To find the time at which cheesecake is being upchucked at the greatest rate, we must take the derivative of the rate which is given to us as T (t). Then we must find the local maximums, which is where T'(t) = 0 and is changing from increasing to decreasing. We must also remember that the maximum could also occur at either of the endpoints at t = 8 or t = 12:

 Therefore, t = 9.184 and t = 10.817. Now these times must be plugged into the original equation to find where the rate was the greatest. Remember to include t = 8 and t = 12:

Therefore, at t = 12, or midnight, is when Jackson was emptying the contents of his stomach at the fastest rate.

Questions? Concerns? Not enough Jackson? Contact me at Jonathan352@gmail.com.

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

Cheesecake on the "x" Axis

Now that we have covered the two major concepts of Calculus (derivatives and integrals), we can move on to combining these two to solve problems.

Think of these two things as your cheesecake ingredients. Like cheesecake is made up of cream cheese and eggs, Calculus is built upon these two concepts.

One of the more applicable situations that these can be used in is the relationships between position, velocity, and acceleration:

 A nice graphical representation

 The order goes like this derivative wise:

The derivative of the position gives you velocity, and the derivative of the velocity gives you acceleration.

And the opposite is true of integrals:

The integral of acceleration gives velocity, and the integral of velocity gives position.

These don't have to exclusively apply to position - velocity - acceleration either. This concept can be universally applied to any function of "something".

If you take the derivative of "something", then you'll get the rate of "something", and if you take the derivative of that then you'll get the change in rate of the rate of "something". 

If this is confusing, applying some Cheesecake to this will clear this up:

At the Cheesecake Factory, they make a lot of cheesecake. As a matter of fact, they make such a great amount of cheesecake that the actual amount of cheesecake cannot be counted. Luckily, the rate at which they make cheesecake per hour can be measured:

Where r(t) is the rate of cheesecake created and t is time in hours. Everyday has 24 hours, which is why time is bounded from 0 hours to 24 hours.

(a) Calculate the amount of cheesecake produced per day.

(b) The rate of cheesecake production starts at one point and then begins falling from there, but at some point in the day cheesecake production begins increasing and increases until the end of the day. Find when in the day this happens.

(c) At what time is a total of 234.85 cheesecakes made? Yes, the reason why there's a fraction of a cheesecake is so that your answer is nice and even.

Solutions:

Will be posted later due to my laziness.

If you crazy people out there did not think this problem is up to AP problem standards, don't worry. The next problem will be BASED upon a REAL AP problem, but, there will definitely be cheesecake.

And the Results are In!

In our first poll, a grand total of twelve people participated.

In the beginning there were five: Cheesecake, Tea, Soup, John Adams, and Strom Thurmond.

The delectable desert was a favorite to win (for obvious reasons), but ironically was barely able to scrape up one vote.

A complimentary beverage and a certain racist politician fought bravely, managing to at one point tie with Soup and John Adams, but eventually lost steam. Tea reportedly returned to its home country of England, while Thurmond reportedly had a heart attack and died at the ripe age of 101.

Then there were two...

Our ever present Minister Plenipotentiary and viscous meal fought it out tooth and nail; spoon and bowl. 


But, in the end there could only be one...

Oh Adams... if you only stopped from passing the Alien and Sedition Act...

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!