The Blue Group: Ben Hamilton, Phil Hassey, Jen Roberts
OE#1: Phil and Dan turn Bad
"Hey Phil," Dan said one afternoon, "Let's go steal some apples from Mr. Armbleman's orchard."
"Why would we want to do that?" I asked, "Isn't stealing wrong." Since this event occurred during third grade, when Dan and I were still sweet an innocent, such ethical questions were occasionally raised.
"So we can eat them?" Dan suggested. The answer seemed appropriate, so we had a plan. "On Saturday afternoon, I'll have my sister carry the coded directions to you."
Pre-Calc
Dan's house is found at point D (0,4) and Phil's house is found at point P (3,0) on a grid map of the Hopkinton, MA area. Dan's sister Becky likes using standard form to express the equation of a line. She would like to know what the equation of the line is between Dan and Phil's house that she is going to have to travel. She also needs to know the distance that she will travel.
If the apple orchard is found at point A (-11,-17) what are the distances that Dan and Phil travel to reach the orchard. What is the total perimeter of the triangle PAD?
When we got to the apple orchard we skulked around quietly, not wishing to be caught.
"How come there aren't any apples in the trees?" asked Dan.
"I think they have already been picked," I said, "Maybe they are storing them somewhere else. Lets go near the shed and see if they are over there."
Slowly we walked towards the shed, hiding behind trees on the way. When we got to the shed, I tried to open the door, but it was locked.
"Locked," I said, "We'll have to knock the door down."
I slowly walked backwards in preparation of hurling my small body through the large imposing door. Though it had chains on it, my mind was certain that I'd be able to break it.
"Wait," said Dan, "I found a half filled box of apples by the side of the shed."
Calculus
The bottom left corner of the box is at point (0,0). The top left corner of the box is at (0,4) and the bottom right corner of the box is at (3,0). The box only contains apples in the triangle defined by those three points. Dan and Phil know that there are 9 apples / ft2. Using integration find the area in the box that contains the apples, and then how many apples are in the box.
Now to generalize, in case Dan and Phil come across other boxes of various size with apples on one side, use the top left corner of the box as (0,h) and the bottom right corner of the box as (b,0). Integrate and derive the formula for finding the area of a triangle.
We figured we could get away with taking three apples. With our three apples in hand we ran off. When we were far enough away, Dan and I sat down and ate our apples. I ate one and Dan ate one. Then we looked at the third.
"I'm bigger, so I get to eat it," said Dan.
I tried to argue, but I gave in. Dan took a large bite out of one side of the apple. He screamed. Probably because of the three live worms that greeted him upon his bite.
"There are worms in this apple!" cried Dan as he threw it in the air.
"Gross," I said, "But this is a valuable asset. Let's keep it for Sunday."
Sunday came soon enough. After Sunday School Dan and I lurked up into the balcony and we sat down. Church began with Pastor Germaine's blessing.
I whispered to Dan, "The apple has gone kind of bad."
"All the better for Uncle Joe's balding head," he said.
We peered over the balcony.
Post-Calc
Dan and Phil notice that the pulpit is at point (0,4). Dan's grandma is sitting at point (3,0) and the door where a greeter stands is at point (0,0). Dan and Phil knew that it would be bad if when the apple splattered on Uncle Joe's head it hit any of those people, because they could run the fastest and catch them. Using a compass and strait edge construct the circle on which those three points lie on. Locate the center point and see if that's where Uncle Joe's head is.
Fortunately Uncle Joe was sitting at the origin of this circle, so upon hitting the target we jumped in celebration. Then we ran. It was all we could do now, before the situation turned really bad.
