Things have been chugging along quite well in Algebra-Land. Even the dreaded, evil, hateful story problems haven’t been so bad.
Until Chapter 4.2 – Finding Equations of Straight Lines. I understand the principles underlying it all, really I do: y = mx + b (slope-intercept form) and (y1 – y2) = m(x1 – x2) (point-slope form.)
But the story problems. These story problems befuddle and vex me. Two examples, if I may:
“According to the U.S. Department of Commerce, there were 24 million homes with computers in 1991. The average rate of growth in computers in homes was expected to increase by 2.4 million homes per year through 2005. Write a linear equation for the number of computers in homes in terms of the year. Let x = 90 represent 1990. Use your equation to find the number of computers expected to be in homes in 2004.”
I get that the slope is 2.4 million. After much exasperated explanation by my charming tablemate, I get that “Let x = 90 represent 1990” is just an off-handed way of saying “let x = 91 represent 1991” and “let x = 104 represent 2004.” I would probably, eventually have figured that out on my own. The rest? Not so much. I now know that (91, 24) is a point. I have a point, I have the slope, thus point-slope blah blah blah. I would never have figured that out. Probably ever.
Here’s one that I haven’t tackled yet. For my own edification, I want to write down my thoughts processes as I’m attacking the problem (perhaps “tentatively poking at” is a better word than “attacking,” which implies more confidence.) Here’s the question:
“Whales, dolphins and porpoises communicate using high-pitched sounds that travel through the water. The speed at which the sound travels depends on many factors, one of which is the depth of the water. At approximately 1000 meters below sea level, the speed of sound is 1480 meters per second. Below 1000 meters, the speed of sound increases at a constant rate of 0.017 meters per second for each additional meter below 1000 meters. Write a linear equation for the speed of sound in terms of the number of meters below sea level. Use your equation to approximate the speed of sound 2500 meters below seal level. Round to the nearest meter per second.”
First thought is, obviously, “oh shit.” Second is, “Clearly, the answer is, ‘I think I’ll shoot myself in the head,'” with all due respect to Hunter S. Thompson via Rigger.
I see that my slope is + .017. I think I have a pair of (1480, 1000), but it might be the other way around, and I have no idea which one I should put in the x position and which in the y. But ok, running with this setup, y – 1000 = .017(x – 1480). Therefore, y = .017x + 1025.16. Plugging in 2500 for x, we get: 1067.66. Ok, but 1067.66 of what? Meters per second. That doesn’t make any sense whatsoever, because for each meter below 1000, the speed increases, not decreases. So perhaps I got my x and y backwards after all. Let’s try it the other way ’round. That gives us an equation of y = .017x + 1463, yielding 1505.5 meters per second, which rounds to 1506 meters per second. That seems right. Yup, the book agrees.
Couldn’t I have just gotten that by doing this: 2500 – 1000 = 1500. 1500 x .017 = 25.5. 25.5 + 1480 = 1505.5 = 1506?
How do I determine which is the x value and which is the y? GAH!
I don’t plan to do terribly well next exam.
On the plus side, I accidentally made everyone in class giggle today. We’d moved on to Linear Inequalities in Two Variables, and our instructor, the intrepid Kary, carefully explained that one needs to shade the area above or below the line to indicate that the answer could be any real number in that range. She mentioned being careful to shade the area in meticulously, and not to be silly about it. “Rats,” I said quietly as I made the universal ‘rats!’ gesture; “I was going to fill it in with puppies.” Evidently, I didn’t speak as quietly as I’d thought. 🙂
One of the student volunteers at the hospital is a math major. He is a senior. He’s freaking out because in one of his current classes, they have begun moving imaginary objects in more than three dimensions. WHAT?! How is that even possible?! Naturally, I had to think of ￼therrin when I heard this, and postulated that such things might be useful in astronomy… maybe… but I can envision story problems from hell, such as:
“The planet Flox, orbiting a dual-sun system in an alternate universe, wants to produce more gleibules. The only way they can increase production without violating the laws of physics is to move their production into the seventh dimension, which, as we all know, subverts the law of gravity. However, operations in the seventh dimension cause an exponential increase in Gesundheit particle emissions, which eventually leak through the trans-dimensional barrier, causing uncontrollable sneezing on Flox.
“The trans-dimensional barrier is represented by the equation, 653 – 456,094,345,001.002353 / 43,980,653xy(z) (insert the rest of a really complicated equation here, involving square roots and exponents and the speed of light.) Assume that Flox experiences time in the same manner as our Earth (this one, right here, where this exam that you are reading right now is located.) Let q = Pelicans. How many gleibules can Flox safely produce in the seventh dimension without necessitating an increase in allergy medications? Should they be concerned about pollutants leaking into the eighth dimension? If so, do their politicians care? Describe the eco-political environment on Flox using only numbers and the word, ‘pants.’ You have 15 minutes.”