Originally posted by: ColdFusion718
Originally posted by: DrPizza
coldfusion - I agree completely with Legendary. I don't know what your problem with his solution is... Furthermore, the OP had already thanked him for the help.
Note the word "HELP." The OP didn't ask someone to do the problem for him, just to help him with it. The relationship necessary to solve the problem is, of course, the pythagorean theorem, which 99% of the time is written in terms of a,b, and c. The only real difference between your solution and his is the substitution of x and y for a and b. Forcing all problems to be in terms of x & y is laziness and reduces a students ability to deal with relationships with other variables. I do not, though, disagree with kyteland's picture that he provided - a picture is worth 1000 words, and lacking a picture, a^2 + b^2 = c^2, take the derivative with respect to t and substitute known values is probably the briefest way to explain the problem. There's absolutely nothing wrong with his inclusion that dc/dt = 0, in fact, IMHO (the opinion of a calculus teacher), dealing with three "variables" now with this problem is probably a better stepping stone to other typical related rates problems, especially those with more than 2 variables, ex. sand falling into a conical pile where V = 1/3*pi*r^2*h. Would you advocate switching the r to x and h to y?? I certainly hope not.
I am an engineer; when I started to do this problem, I skipped a few steps in my head. I gave the OP the benefit of the doubt in that he is smart enough to know that when you stick a ladder against a wall, it always forms a triangle with the ground; hence, the Pythagorean Theorem comes to mind. Moreover, I used X and Y for the horizontal and vertical distances, respectively, because they reduce confusion when one goes back to substitute in values--most students are trained to immediately know that X = horizontal and Y= vertical.
I was aiming for efficiency and reduction in probability of making errors during substitution of values at the end. If the problem were stated with a pre-defined coordinate system, then one would adjust one's brain accordingly to accommodate.
With regards to your conical sand example, one would not make the following substitution: r = x; h = y. Why not in this case? You're dealing with a 3D problem, whereas the previous problem you were dealing with a 2D problem which
begs the student to use X, and Y because it is the most logical thing to do.
Yes, sometimes certain teachers flip it around to try and hinder students, which brings to mind a side-note: If a student already understands how to do a problem and can show said understanding with a diagram, equations, and a correct solution on an exam, why does a teacher feel the need to put unnecessary tricks to hinder the student? Afterall, I assume that the teacher is testing the student's understanding of the concepts and differentiation skills, not bookkeeping or, to put it in colloquial terms, "how well you not fvck up on the exam." DrPizza, if you're one of those Calculus teachers, I'd say, with all due respect, "Fvck you, sir."
I discovered quite a few years ago that many students are intimidated by variables other than x & y. I think part of this is the fault of generic math books that while pretending to have real world application problems, really spend too much time manipulating x's and y's in problems. I find it to be quite a handicapping problem for many of these students. This is most recognizable in physics students who have completed 3 years of high school math, yet cannot identify the shape of the graph of KE=1/2mv^2, with KE on the vertical axis and velocity on the horizontal axis. Not only do they have problems when faced with other variables (especially any greek characters), but they freeze when x is on the vertical axis (ex. elongation of a spring as a function of time). It's not the physics that hinders them, it's the lack of proper mathematical preparation that hinders some of their learning.
Most importantly, people in your little world of x's and y's fail to see many of the connections between the real world and mathematics, particularly science and mathematics - mathematics is completely abstract to them. In my state (NY), students begin learning the pythagorean theorem in 7th grade, before they learn x and y. Thus, I don't see why it's more logical to use x & y rather than a,b, and c. Ironically, the benefit of the doubt you gave the OP - that you use the pythagorean theorem - is probably the only clue 95% of students need when stuck on this problem. They'll hear "pythagorean theorem" and say, "got it," usually with that look on their face that says they realize how easy that problem is now and can't believe they didn't think of it on their own.
Nonetheless, I still believe that x & y are (generally) abstract variables and over-emphasis and use of them results in less understanding by the students (although they can still breeze through the standard types of questions given on exams such as the AP). My first year teaching calculus, I could give the problem y = x^3 + 7x^2 - 8x + 5 and have students differentiate with respect to y, but then if I gave them p = r^3 + 7r^2 - 8r + 5, and told them to differentiate with respect to p, they'd have trouble and take 5 times as long to do the problem. Since then, I've made an effort starting in pre-calculus to get them to realize that they're just variables... it doesn't matter what letter they are or from what alphabet they came from.
When a student can't differentiate with respect to anything but y or t, there's a problem. Apparently you disagree and want me to go back to x's and y's, because otherwise there would be "unneccesary tricks to hinder the student." The truth is, it's sticking to x's and y's that hinders the students from mastering the mathematics and the ability to operate at a higher cognitive level. I keep my expectations high for my students. (especially since I have the cream of the crop students taking calculus.) 3 weeks before this year's final exam, I pulled a half dozen calc I finals from various universities off the internet. I asked my students to look at the tests and a few admitted they'd be insulted if their final exam was as easy.
No offense meant to engineers as a whole, but in my experiences (3 1/2 years of ceramic engineering (includes chemical, mechanical, and electrical engineering), a total of 8 1/2 years of undergraduate and graduate work along side engineers) engineers and engineering students often "think" they're smart, but in actuality they're just trained to do the problems "just like in the book" - or to put it another way, they're "book smart" but aren't as good at applying that knowledge. One example that stands out from my undergrad math courses is the differential equations course I took at Pitt - 2 math majors (the other one was *hot*, I wish I had pics of her, we were inseparable for 2 years but I never took pics) and the rest of the class was engineering students. I loved the course because it wasn't "here's how you do this," rather, we were taught how to understand the equations, attractors, repellers, vector fields, etc. The course had minimal emphasis on "how to solve a 1st order linear differential equation," etc. Karen and I easily had 100 averages. The grades posted at the end of the semester were A+, A+, and none higher than a C+ after that - more evidence to me that many of the students who go into engineering are often "book smart" but not as capable as others at actually understanding what they're doing. (or at least, not as capable as the math majors) I'm not saying that all engineers are this way - I've known a few incredibly bright and capable engineers, but in my experiences it's far from the majority. However, the majority of people with "engineer" as part of their job title seem to feel that they automatically qualify as "bright." My apologies to any engineers who feel insulted by this (but then again, some of you probably recognize this problem as well as I did when I was in engineering.) Hopefully, none of my students who enter engineering won't have the same handicap as others, and hopefully they won't have a big chip on their shoulder.
X=12 and dX/dt = 2, then t = 6. If t =6 and dY/dt = -4.8, then the ladder ends up underground
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