I am having trouble identifying whether a problem is a linear growth problem or an exponential growth problem - page 153 of the MGMAT algebra book:
Jake was 4.5 feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt? (12 inches = 1 foot)
I understand how to approach linear growth problems and exponential growth problems, so my question does not deal with the actual math. I am more concerned with the wording of the problem. It says that Jake "grew at a constant rate." When I read this, I instantly think that this means that he grew at a constant "percentage" each year because isn't a rate a percentage? A rate is not a nominal amount, right? When I read this question, I want to use the exponential growth approach. If the problem said, "Jake grew at a constant amount..." then it would be very clear to me that this is a linear growth problem. What are some of the key words that GMAC uses, so I don't get confused on the test?
Taking it on Saturday (6/1), so I appreciate your timely response!
Thanks!
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- c.w.richardjr
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Re: Linear Growth vs. Exponential Growth
a "constant rate" is, indeed, what you are calling a "constant nominal amount".
although you shouldn't need the justification for practical reasons (= you should just know this, if you're ever called upon to use it), here's the justification, if you care:
"rate" is defined as (unit of measure) PER (other unit of measure). that's the definition of a "rate" in mathematics"”whether it's miles per hour, or feet per second, or inches per year, or gallons per worker, or blah blah blah xxxxx.
so, a "constant rate" refers to a constant number of miles per hour, or a constant number of feet per second, or a constant number of inches per year, or whatever.
if you had growth by a fixed percentage, that would be "growth by a constant factor", not "growth by a constant rate".
(if you are thinking about interest rates here, then that's a historical anomaly. originally, "interest rate" referred to simple interest, in which the interest is compounded using only the original principal regardless of how much time had passed, and was thus a true "rate". but for compound interest, it's not really a "rate" anymore"”that's just the word that people use because, well, it's the word they always used before.
just like you can "write papers" entirely on a computer these days... but are they really "papers"? well, yes and no. eh.)
2 more comments here:
1/
in this particular problem, there's no issue of ambiguity"”you may not have been reading the problem carefully enough.
the question part says...
how many inches per year did Jake grow during his growth spurt?
... thus implying that the number of inches per year was constant during the growth spurt.
if "constant rate" referred to a constant percentage, then this question would be complete nonsense.
2/
even if you have this issue on the exam, rest assured that (a) if you use the wrong interpretation then the numbers will just be horrible beyond belief, AND (b) there won't be a wrong answer corresponding to the incorrect interpretation of the words.
although you shouldn't need the justification for practical reasons (= you should just know this, if you're ever called upon to use it), here's the justification, if you care:
"rate" is defined as (unit of measure) PER (other unit of measure). that's the definition of a "rate" in mathematics"”whether it's miles per hour, or feet per second, or inches per year, or gallons per worker, or blah blah blah xxxxx.
so, a "constant rate" refers to a constant number of miles per hour, or a constant number of feet per second, or a constant number of inches per year, or whatever.
if you had growth by a fixed percentage, that would be "growth by a constant factor", not "growth by a constant rate".
(if you are thinking about interest rates here, then that's a historical anomaly. originally, "interest rate" referred to simple interest, in which the interest is compounded using only the original principal regardless of how much time had passed, and was thus a true "rate". but for compound interest, it's not really a "rate" anymore"”that's just the word that people use because, well, it's the word they always used before.
just like you can "write papers" entirely on a computer these days... but are they really "papers"? well, yes and no. eh.)
2 more comments here:
1/
in this particular problem, there's no issue of ambiguity"”you may not have been reading the problem carefully enough.
the question part says...
how many inches per year did Jake grow during his growth spurt?
... thus implying that the number of inches per year was constant during the growth spurt.
if "constant rate" referred to a constant percentage, then this question would be complete nonsense.
2/
even if you have this issue on the exam, rest assured that (a) if you use the wrong interpretation then the numbers will just be horrible beyond belief, AND (b) there won't be a wrong answer corresponding to the incorrect interpretation of the words.
2 posts Page 1 of 1