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October 8th, 2017, 12:44 AM   #1
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I am stuck in this problem about math and chemistry marks.

(ex) Suppose that you have math and chemistry exams on the same day. You have 20 hours of study time remaining. You estimate that your mark on the chemistry exam as a function of hours spent studying is

C(x)=100x/(4+x)

You have a similar function for math defined below.

M(x)=100x/(k+x) where k > 0.

(a) Suppose that k > 4
Interpret what this means about the comparative difficulty of studying math and chemistry.

(b) Without doing any calculations, do you think you should spend more time studying for math or chemistry?

(c) Suppose that k > 0
let A(x) be a function for your average mark on the 2 exams where x is the number of hours spent studying math. Assume that the remaining time is spent entirely on chemistry. Write your answer in terms of the functions C and M.

Find the critical points x1 and x2. For what values of k is x1 in the domain of the model? For what values of k is x2 in the domain of the model?

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my solution
(a) if 0<k<4, C(x)<M(x) meaning chemistry is easier since it takes less time to study.
if k>4, C(x)>M(x) meaning chemistry is harder since it takes more time to study.

(b) I would spend more time studying for math since k varies on the intervals
as indicated in (a).

(c) I define A(x) as follows: A(x)=(M(x)+C(x))/2.

Differentiating the function and setting the derivative to 0 give two critical values

x1=2*sqrt(k)*(12-sqrt(k))/(sqrt(k)+2)

x2=2*sqrt(k)*(12+sqrt(k))/(sqrt(k)-2)

In x1, I set 12-sqrt(k) > or = 0 and solve it to get k < or = 144.
In x2, I set sqrt(k)-2 > or = 0 and solve it to get k > or = 4.

Then, combining the inequalities gives 4 < or = x < or = 144.

Now, I am stuck. I don't know how to answer part c.

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I am a math tutor. I am trying to help my students prepare for a math test. But, I am struggling with the problem.

Could someone please help me? Thank you very much for your help.
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October 8th, 2017, 03:48 AM   #2
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For (a), you are told to suppose that k > 4. This means C(x) > M(x), so it takes less study time to achieve a particular mark in chemistry than to achieve the same mark in mathematics.

That isn't quite the same thing as asserting that studying mathematics is more difficult than studying chemistry - just that it takes more time studying to achieve the same mark.

(b) One should spend more time studying mathematics if the chemistry and mathematics marks are equally important and k > 4. If 0 < k < 4, the opposite conclusion would apply. If k = 4, more time should be spent on the subject considered to be more important, or the same time should be spent on each if they are considered to be equally important. What happens if k isn't positive?

(c) A(x) = (M(x) + C(20 - x))/2 = (100x/(k + x) + 100(20 - x)/(24 - x))/2.
When is the derivative of that equal to zero?
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October 8th, 2017, 03:55 AM   #3
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For (a), you are told to suppose that k > 4. This means C(x) > M(x), so it takes less study time to achieve a particular mark in chemistry than to achieve the same mark in mathematics.

That isn't quite the same thing as asserting that studying mathematics is more difficult than studying chemistry - just that it takes more studying to achieve the same mark.

(b) One should spend more time studying mathematics if the chemistry and mathematics marks are equally important and k > 4. If 0 < k < 4, the opposite conclusion would apply. If k = 4, more time should be spent on the subject considered to be more important, or the same time should be spent on each if they are considered to be equally important. What happens if k isn't positive?

(c) A(x) = (M(x) + C(20 - x))/2 = (100x/(k + x) + 100(20 - x)/(24 - x))/2.
When is the derivative of that equal to zero?
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October 8th, 2017, 04:56 AM   #4
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"C(x)" is the mark you will get in chemistry if you study chemistry for x hours.
"M(x)" is the mark you will get in math if you study math for x hours.

"C(x)> M(x)" means that you will get a higher grade in chemistry than in math for studying the same amount of hours, not that you need to study more hours. This says that Chemistry is easier than math.
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