Isolating Groceries from Total Expenses, and Problems from Paragraphs
Date: 09/03/2011 at 20:18:01 From: Wendi Subject: R(t)=T(t)-G(t) Average monthly expenditures for the Mays family's groceries and rent has increased year after year. The total amount spent on groceries and rent, combined, can be estimated using T(t) = 0.1x^2 + 49x + 1125 Here, T(t) is the average monthly amount in dollars, and t is the number of years since 1995. The amount spent on groceries alone can be estimated using G(t) = 1.1x^2 + 2x + 465 Here, G(t) is the average number of dollars spent per month for groceries, and t is again the number of years since 1995. Write an equation for the average monthly rent expense R(t) = T(t) = G(t). Choose the right answer: A) R(t) = x^2 - 47x - 660 B) R(t) = x^2 - 47x_660 C) R(t) = 1.1x^2 + 47x + 660 D) R(t) = x^2_47x + 660 This whole thing is confusing to me. I am now taking college classes after 26 years out of school.
Date: 09/03/2011 at 23:20:26 From: Doctor Peterson Subject: Re: R(t)=T(t)-G(t) Hi, Wendi. I've worked with lots of students in your position, and believe me, I appreciate the effort it takes! This is the sort of problem that uses a long story to disguise what is really a simple problem. Sometimes what you have to do, at least in your mind, is paraphrase the problem with all the extraneous details removed. Replacing x with t (as the original problem stated, I'm sure), and changing the last "=" to "-" (a typo): T(t) = 0.1t^2 + 49t + 1125 Here, T(t) is the average monthly amount in dollars, and t is the number of years since 1995. G(t) = 1.1t^2 + 2t + 465 Here, G(t) is the average number of dollars spent per month for groceries, and t is again the number of years since 1995. Write an equation for the average monthly rent expense, R(t) = T(t) - G(t) In fact, we can do without the definitions of the variables and the meaning of the functions: T(t) = 0.1t^2 + 49t + 1125 G(t) = 1.1t^2 + 2t + 465 R(t) = T(t) - G(t) They've just defined two functions, and asked you to subtract one from the other. You have to subtract because of the meaning: the total is groceries plus rent, so the rent is the total minus groceries. The main purpose of this problem is to get you used to function notation. Subtracting two functions just means subtracting the expressions they represent, so what they're asking for is something you've probably done many times: Simplify the expression (0.1t^2 + 49t + 1125) - (1.1t^2 + 2t + 465) (By the way, the correct answer is not in your list, but you've got more typos there anyway.) Again, although now you should have no trouble completing the problem, you'll want to focus on how to extract the real problem from all the "noise" to get to this point. One way is to read backward (after reading the problem through the usual way, of course). That is, look for what they are asking (here, an expression for the function R); then for how R is defined (as the difference of two functions); and in turn, for how those functions are defined (by expressions). That allows you to focus on what is really needed, and bypass the trivia. In a "real" problem, of course, they wouldn't have given you the expressions; you'd have had to work them out from the meanings of all the words. But in this problem, there's not enough information to have done that, anyway. Finally -- unless the book did this, which would be unforgivable -- your typos suggest you may need help on one more point. You changed many of the t's of the expressions to x's. Possibly you are not yet quite comfortable with an essential idea in function notation: the variable in the expression for a function is not always x, but is whatever is inside the parentheses. Writing T(t) tells you that the variable they are using is t, so that when you evaluate T(1), say, you'd replace t with 1. This t is a placeholder; any letter could have been used just as well. You may fully understand this, and have typed the x's purely out of finger-habit, but I thought I'd mention it in case it's more than your fingers that need reminding! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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