MTH 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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I have a big problem with question 8 in the rates qa. In the next field I will paste the entire problem from the question down to the self-critique score. I really think that the given solution is wrong and if you read my self-critique you will see exactly why.
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Question: `q008. In a study of how lifting strength is influenced by various ways of training, a study group was divided into 2 subgroups of equally matched individuals. The first group did 10 pushups per day for a year and the second group did 50 pushups per day for year. At the end of the year to lifting strength of the first group averaged 147 pounds, while that of the second group averaged 162 pounds. At what average rate did lifting strength increase per daily pushup?
Your solution: This is worded ambiguously because we can't find the average rate of increase in lifting strength per pushup since we don't know the initial lifting strengths. All we can compare is how much more strength per extra pushup did group 2 gain over group 1. So group 2 did 40 extra pushups per day and they gained an extra 15 lbs. of lifting strength over the year because of it. So we need to know how much their strength increased on average per day which is 15 lbs./ 365 days or 0.0411 lbs/day. Now we know they did 40 extra pushups per day but we need to use the opposite of this, that is 1 day/ 40 pushups then with some dimensional analysis we multiply 1 d/40 pushups by 0.0411 lbs/ day to get lbs increased per pushup which is 1.028x10^-3 lbs/pushup.
Your argument is very well presented, and you clearly understand what you're doing here.
What you have calculated is the average rate at which strength increases per daily pushup, per day.
The question asked for the average rate of strength per daily pushup, which is different.
But note the following:
On a graph of lifting strength vs. number of daily pushups, the two known points would be (10, 147) and (50, 162). The 'rise' between these points would be 15, corresponding to 15 pounds of increased lifting strength. The 'run' would be 40, corresponding to 40 daily pushups. The slope would therefore be .375, corresponding to an average rate of change of .375 pounds per daily pushup.
This slope is the average rate of change of lifting strength with respect to the number of daily pushups.
Further notes, since you have already completed your first-semester calculus course:
If the graph is a smooth curve (i.e., differentiable), then the slope at a point represents the instantaneous rate of change of lifting strength with respect to number of daily pushups.
Under the same conditions, the mean value theorem guarantees that there exists a point within the interval at which the derivative is equal to the average rate of change. We would be guaranteed that there is at least one point at which the instantanous rate of change is .375 lbs / pushup.
In an application it might or might not be important to take the time frame into account, in which case the calculation you did, the average rate at which strength increases per daily pushup, per day, is relevant. Or, equivalently, the average rate at which strength increases per daily pushup, per year (which would be .375 lbs / daily pushup / year).
Whether or not the question as stated is in fact ambiguous isn't particularly important, and if you want to disagree on that point it's no problem, since as I said you obviously understand this situation very well.
The following version of the intended statement is not ambiguous. It's a little too formal to use on first-semester students:
The average rate of change of strength with respect to number of daily pushups is (change in strength) / (change in number of daily pushups), which is 15 lbs / (40 daily pushups) = .375 lbs / daily pushup.
Confidence Assessment: 3
Given Solution:
The second group had 15 pounds more lifting strength as a result of doing 40 more daily pushups than the first. The desired rate is therefore 15 pounds / 40 pushups = .375 pounds / pushup.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
STUDENT COMMENT:
I have a question with respect as to how the question is interpreted. I used the interpretation given in the solution
to question 008 to rephrase the question in 009, but I do not see how this is the correct interpretation of the question as
stated.
INSTRUCTOR RESPONSE:
This exercise is designed to both see what you understand about rates, and to challenge your understanding a bit with concepts that aren't always familiar to students, despite their having completed the necessary prerequisite courses.
The meaning of the rate of change of one quantity with respect to another is of central importance in the application of mathematics. This might well be your first encounter with this particular phrasing, so it might well be unfamiliar to you, but it is important, unambiguous and universal.
You've taken the first step, which is to correctly apply the wording of the preceding example to the present question.
You'll have ample opportunity in your course to get used to this terminology, and plenty of reinforcement.
Self-critique (if necessary): The given solution cannot be right because they did 40 extra pushups per day over the whole year not just 40 extra pushups over the whole year. This becomes clearer if we look at total extra pushups for the year which is 40 pushups/day over 365 days which is 14600 extra pushups. If they gained .375 lbs/pushup as the solution says they would gain a total of 14600 pushups * .375 lbs/pushup to get 5475 extra lbs. This is obviously not right. My solution however works out just right because when you multiply the 1.028x10^-3 lbs/pushup by the total pushups, 14600, you get 15 lbs which is the total gain that group 2 had over group 1.
Self-critique rating #$&* 3
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As you can see I don't think there is any way the answer is .375 lbs/pushup because when you multiply this by the actual total number of extra pushups they did over the entire year the number is way more than any human could ever lift. Doesn't this make sense?
&#Good work. Let me know if you have questions. &#