Robofootball: Doublespin

Back at the Tucker home for another tutoring session, they sat in her spacious bedroom; she had a massive desk with a 15-square foot surface top since it measured 60 inches in length by 36 inches in width. There was plenty of room for two, maybe even three despite her two computer monitors, printer, keyboards, and purses. She had two D & B purses along with one marked Coach.

“Okay Jess, here’s your first story problem, a train leaves Seattle at 9:00 a.m. Pacific Time while another one leaves Chicago at 11:00 a.m. Central Time. The Seattle train averaged 40 miles per hour while the Chicago one goes 50 mph. The distance between the two stations is 2,000 miles. If the trains are traveling toward each other on the same track, what time (Mountain Time) will they crash into each other? As an extra bonus, approximately where will they crash?”

“Well one thing he made easy for you,” Carly went on, is that they are both leaving at the same time.”

“Huh?”

“Come on Jess, that’s easy, Pacific Time is an hour behind Mountain Time and Central is an hour ahead.”

“Oh I see, they both leave at 10 a.m.”

“Mountain Time.”

“Right.”

“You see Jess it’s a fairly simple 2 variable, 2 equation problem. Make the little rate x time = distance box and put the numbers and letters in.”

“Huh?”

“Wait, now that I think of it, it’s easier than I thought, we can probably just graph it and get an intersection point, I mean you can logic this one out.” Carly paused and looked at his confused face. He had his head in his hands. “What’s the matter Jess? You’re usually a litter quicker with this stuff. The ACT retakes are next Saturday.” It was Sunday afternoon and an hour yet before the Tucker Sunday dinner.

“Just tired I guess, still trying to work 20 hours a week mostly on weekends, go to school too, play football, ……, you know.”

“Are you sure you’re okay? That was a rough game in the rain Friday night.”

“Sure, no problem,” but he didn’t let on the frequency and severity of the headaches he had been suffering constantly since game 2. Now they were feeling like migraines after the hit he had taken Friday night.

“I’m not an expert, but I think you suffered a concussion no matter how the doctors sugar coat it.”

“There’s only one more game, well, not counting the playoffs.”

Carly sighed, “I know your potential scholarship offers depend on it I’m sure.” It was a Catch-22 in her mind. If he didn’t play, a key scout could make the difference in what or what not was offered. If he did play, he could jeopardize his health. Then again, the way the national hearings were going on football, the sport could be finished.

“I did get that letter from the University of Michigan.”

“Oh?”

“Yeah, they want to schedule a recruiting appointment after the season is over.”

“I guess that’s great news Jess,” she recalled viewing the large stack of mail from potential colleges for him that was twice the size of her own despite her academic accomplishments.

“Maybe we can go together, won’t that be great?”

“Have you heard anything?”

“Not from them, but it’ll probably depend on my ACT scores. I averaged 31. 5 last year out of 35, I want to ace it this year.”

“I think I only had 24.”

That’s probably good enough to get you in with sports, but we do get to take it again as seniors. My mom told me that in her day, you only get to take it once. Anyway, let’s get back to your rate-time-distance problems. I see Mr. Pritchett is at it again with crashing trains!”

“How so?”

She looked at Jess’s confused face. “Really Jess, if the Seattle train is going 40 mph, how long will it take to go 2,000 miles?”

“Oh, I guess that’s easy,” he divided 40 into 2,000 and came up with 50 hours. In a bit of inspiration he divided 50 into 2,000 and came up with 40 hours for the Chicago train. “Can’t we just take half the 50 and half the 40, and average it out?” He paused and came up with 22 ½.

“Well your logic is somewhat sound, but it’s really a rate problem, and that doesn’t account for the extra speed in Chicago. See, I came up with 22.2 repeating or 22 2/9.”

“How’d you get that?”

“Well, with simple rates, you can get by with one variable, x over 40 for the Seattle train and x over 50 for the Chicago one.”

“Shouldn’t we set it equal to 2,000?”

“No, it’s like the lawn mower problem we did last week, think of it as 1 job or 1 trip. You know, you can cut the grass in 1 hour, your rate is x over 1. If I can do it in 2 hours, then it would be x over 2. Together x/1 + x/2 would equal 1 for 1 job.”

“So would if you split the difference, wouldn’t it only take us ¾ of an hour working together?”

“No, that’s where you’re slightly off, work out x/1 + x/2 = 1 and see what you get.”

“x/2 is the same as ½ x, right?”

“Yes.”

“So together it’s 1 ½ x.”

“Well, sort of.”

“What do you mean?”

“I remember Mr. Pritchett saying that mixed number constants in front of variables is sloppy, better to use 3/2 x.”

“Okay, 3/2 x = 1, you multiply by the reciprocal right?”

“Yes Jess, you should remember that from Algebra I.”

“Let’s see, that’s just 2/3, and 2/3 times 1 is still 2/3.”

“Well there’s your answer,” said Carly. “It should take us 2/3 of an hour working together, not ¾. You know, you’d probably be okay on the ACT with the multiple choice format thinking about it your way. See what I did for the train problem, x/40 + x/50 = 1?” She had actually computed the 22 2/9 in her head from there.

“Common denominator is 40 x 50 or 2,000,” said Jess.

“Sure, you use 200 which is the LCM.”

“Huh?”

“Never mind, 2,000 works, go ahead and work it out.

Jess did so successfully and got 50x/2000 + 40x/2000 = 1, then 90x/2000 = 1, followed by x = 2,000/9 reduced to 22 2/9 like Carly’s answer.

“That’s good Jess, we’ll have to check the atlas. At 22 2/9, the Chicago train goes about 1,111 miles and the Seattle train goes about 809 miles.” Jess pulled a ruler out of Carly’s top desk drawer while she pulled out a National Geographic Atlas off of her bookshelf.

“Tricky,” said Jess. “Looks like it’s about on the corner where South Dakota, Montana, and Wyoming all meet.”

“Carly laughed, “Mr. Pritchett is always doing stuff like that. You should get an extra bonus if you write that up. Now, let’s actually do one with the d = rt formula. The airplane ones are easier, you just do x + y for the rate when you have a tail wind, and x – y against the wind, and put your numbers in for t. Make the two equations with the two variables and I know you know how to solve them after that.”

“Yeah, I suppose,” and he did work one out.

“Plus Mr. Pritchett is pretty cool, he’ll only put one or two harder story problems on your test, just to separate the A’s from the B’s he likes to say.”

“I always wondered what this has to do with anything,” said Jess just a little frustrated. He was still having trouble focusing and was making an effort to do the math and hide his problems from Carly simultaneously. As a result, the pounding in his head was starting to feel like a jackhammer on some stubborn cement.

“Well, Mr. Pritchett always told us that math is more right-side brain, you know, artsy, imagination, and deep thinking; but there’s that logical problem-solving aspect too. It helps with deductive and analytical reasoning, makes you a better thinker. Plus once you go to college, Mr. Pritchett indicated that to be considered worldly and well educated, one should we well rounded and somewhat versed in a wide variety of subjects.” Carly looked back at Jess who was staring off vacantly. “Sorry Jess, but if you think this stuff is hard, you should see some of the extra bonus work he gives us in AP Calculus.”

“I still don’t see what it’s good for.” He was asking the wrong person.

“Well here, let me show you a couple of examples. Say you have a hundred feet of fence and want to maximize the area.” She drew a rectangle 10 feet by 40 feet and a square 25 feet by 25 feet. “What’s the perimeter of the two?”

Jess paused a moment, “100 feet.”

“Right, now what’s the area?”

“Let’s see, length x width,” Jess grabbed his calculator, “The rectangle is 400 feet.”

“That’s 400 square feet, two dimensions for area. You know when you times by 10 you just have to add a zero to the other number.”

“Whatever.”

“Now do the other rectangle.”

“Isn’t it a square?”

“A square is just a special version of a rectangle where all the sides have the same length. It’s still a 4-sided figure with 90 degree angles.”

“Whatever,” Jess punched 25 x 25 and came up with 625, “625 SQUARE FEET,” he answered.

“Okay, see the point here?”

“Yeah, we used the same amount of fence but got more area with the square.”

“Exactly!”

“But what does this have to do with Algebra?”

“Bear with me, now what happens if we still have 100 feet of new fence to put up, but we’re going to connect it to an existing fence, what would you do?”

“Probably just make a square again.”

“Okay, try it,” Carly drew a picture for him, “Now how long would you make the sides?”

Jess punched in 100 and divided by 3, “33 point 3 repeating.”

“Or 33 and a third,” said Carly as she put the numbers in her drawing”

Existing fence

+___+___+___+___+___+

| |

33 1/3 | | 33 1/3

| |

---------------------------------

33 1/3

“Use the fractional mode on your graphing calculator and what do you get for the area?”

“10,000 ninths.”

“Which is what?”

“1,111 point 1 repeating.”

“Okay, so about 1,111 square feet.”

“Yup, that’s it, you really don’t need Algebra when you can just make a square every time.”

“Are you sure that’s the best you can do?”

“Uh oh, probably not.”

“No, here, let’s draw it again and use a little Algebra.”

Existing fence

+___+___+___+___+___+

| |

x | | x

| |

---------------------------------

Length

“Let’s call the small sides or the width “x”. If we have 100 feet of fence to work with, what would the length or long side be in terms of x?”

“I don’t know,” said Jess whose headache was gradually reaching migraine status.

“It’s a total problem,” Carly hinted. “Take the total that you have, in this case 100, and subtract out what’s already labeled.”

“100 minus x?”

“Close, you have two x’s.”

“Oh, 100 – 2x?”

“Yes, now how do you do the area?”

Existing fence

+___+___+___+___+___+

| |

x | | x

| |

---------------------------------

100 – 2x

“Length times width.”

“Yes, the width is x and the length is 100 – 2x. Go ahead and times it out.”

Jess wrote x(100 – 2x) = 100x - 2x2

“There, you have a quadratic equation with that 2nd power, but let’s put it in the right order,” she wrote -2x2 + 100x.

“Okay, what do we do now?”

“You can graph it for one on your calculator to see what it looks like.”

Jess did so, “Oh yeah, I remember, these are parabolas.”

“Do the maximum, or just look on the table.”

“It’s when x = 25,” Jess clicked through. If there were two things the modern teenager was quite adept at, it was clicking buttons for short cuts, whether it was a computer, texting on a plane, beating on an IPad, or in this case, a graphing calculator.

“Mr. Pritchett always made us do it on paper too, do you remember that in a parabola, the maximum or minimum is x = -b/2a. In quadratic form, the “a” is the number in from of x-squared and “b” is the number in front of x.”

“Yeah, so “a” would be -2 and “b” would be 100.”

“Good, now what’s –b over 2a?”

“Let’s see, Jess punched it in again, it comes out to 25.”

“All right that matches, now go back to the drawing and substitute 25 in for x.”

“Okay.”

“What’s the width?”

“Just 25.”

“Good, what’s the length?”

Jess punched in 100 – 2(25), “50” he said.

“Now what’s the area?”

Existing fence

+___+___+___+___+___+

| |

25 | | 25

| |

---------------------------------

50

“Let’s see, 25 times 50,” Jess said aloud, “It’s 1,250.”

“One thousand two hundred fifty SQUARE FEET,” Carly emphasized.

“Yeah, so?”

“What did you write down when you made it a square?”

“One thousand one hundred eleven and some change.”

“1,250 square feet is bigger.”

“Oh,” was Jess said trying not to rub his aching temples any more.

“There you go, that’s what good Algebra is.”

“Okay, but I don’t know if it’s really my thing.”

“You never know. In Calculus, we’re doing maximums and minimums with 3-dimensional objects like cans, trying to get the most volume while minimizing the amount of material used. Turns out a can whose height is twice the diameter is the most efficient.”

“Oh god,” Jess unconsciously rubbed his temples this time.”

You know, things like Beef-a-roni and canned peaches are good cans, but there’s a lot of waste in say a 6 ½ ounce tuna can. Of course Mr. Pritchett says that a lot of machinery like tuna canneries were set up long ago and it wouldn’t be cost effective to purchase new machinery just to make a more efficient can.” She looked at him as he had now lost total focus. “Sorry Jess, should we get back to some more rate-time-distance problems?”

“I suppose.”

“Kids!” Sarah Tucker shouted from downstairs, “It’s time for dinner!”

“Thank god,” mumbled Jess, any more math and he’d be ready for the Psych Ward.

“Saved by the bell,” Carly smiled. She had been so involved in the math that she failed to notice Jess’s deteriorating condition.

“No man is an island, entire of itself; every man is a piece of the continent, a part of the main; if a clod be washed away by sea, Europe is the less, as well as if a promontory were, as well as if a manor of thy friends or of thine own were; any man’s death diminishes me, because I am involved in mankind; and therefore never send to know for whom the bell tolls; it tolls for thee.”

John Donne, Devotions: No Man is an Island, Ib.XVII