It is the final exam period and you have only math and chemistry left. They are on the same day, and you have only twenty hours of study time remaining. You estimate that your mark C on chemistry, as a function of hours x hours spent studying, is given by:
. . . . .\(\displaystyle C(x)\, =\, 100\, \dfrac{x}{4\, +\, x}\)
You can't be quite as certain about your mark M in your math course, but you know it has a similar form:
. . . . .\(\displaystyle M(x)\, =\, 100\, \dfrac{x}{k\, +\, x}\, \mbox{ for }\, k\, >\, 0\)
1. For this part only, suppose that k > 4. Interpret what this means about the comparative difficulty of studying math and chemistry. Without doing any calculations, do you think you should spend more time studying for math or for chemistry? Give an honest answer based on your thoughts before doing the rest of this OSH. This answer will not be marked for correctness of your guess; it is intended to encourage you to build an expectation that you can compare later to your formal answer.
2. Write down a function for your average mark, A (x ), given by:
. . . . .\(\displaystyle x_1\, =\, \dfrac{2\sqrt{\strut k\, }\,\left(12\, -\, \sqrt{\strut k\,}\right)}{\sqrt{\strut k\,}\, +\, 2}\, \mbox{ and }\, x_2\, =\, \dfrac{2\sqrt{\strut k\, }\,\left(12\, +\, \sqrt{\strut k\,}\right)}{\sqrt{\strut k\,}\, 1\, 2}\)
3. What is the domain of the model? That is, what is the set of x -values that can be plugged into A (x ) in both a mathematical and realistic sense, given the meaning of x in the problem?
4. Verify, by calculation, that there are two critical points of A (x ) given by x1 and x2 (as defined above).
5. Show that any critical point of A (x ) that is on the interior of the domain of the model (i.e., in the domain but not one of the endpoints) is the absolute maximum on the domain.
6. 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? You can use a graphing tool to plot the x1- and x2-values as functions of k. (There is no need to include this graph when you submit your answers.)
7. For what values of k is it best (in the sense of the largest value of A (x )) to spend more time studying math than chemistry? You must show your calculations.
8. Explain in words why this model predicts that it is best (again, in the sense of the largest value of A (x )) to spend the vast majority of your study time on chemistry when math is much easier than chemistry (k << 4) ANDE when math is much harder than chemistry (k >> 4).
Can anyone please help with this questions! Thanks so much =)
. . . . .\(\displaystyle C(x)\, =\, 100\, \dfrac{x}{4\, +\, x}\)
You can't be quite as certain about your mark M in your math course, but you know it has a similar form:
. . . . .\(\displaystyle M(x)\, =\, 100\, \dfrac{x}{k\, +\, x}\, \mbox{ for }\, k\, >\, 0\)
1. For this part only, suppose that k > 4. Interpret what this means about the comparative difficulty of studying math and chemistry. Without doing any calculations, do you think you should spend more time studying for math or for chemistry? Give an honest answer based on your thoughts before doing the rest of this OSH. This answer will not be marked for correctness of your guess; it is intended to encourage you to build an expectation that you can compare later to your formal answer.
2. Write down a function for your average mark, A (x ), given by:
. . . . .\(\displaystyle x_1\, =\, \dfrac{2\sqrt{\strut k\, }\,\left(12\, -\, \sqrt{\strut k\,}\right)}{\sqrt{\strut k\,}\, +\, 2}\, \mbox{ and }\, x_2\, =\, \dfrac{2\sqrt{\strut k\, }\,\left(12\, +\, \sqrt{\strut k\,}\right)}{\sqrt{\strut k\,}\, 1\, 2}\)
3. What is the domain of the model? That is, what is the set of x -values that can be plugged into A (x ) in both a mathematical and realistic sense, given the meaning of x in the problem?
4. Verify, by calculation, that there are two critical points of A (x ) given by x1 and x2 (as defined above).
5. Show that any critical point of A (x ) that is on the interior of the domain of the model (i.e., in the domain but not one of the endpoints) is the absolute maximum on the domain.
6. 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? You can use a graphing tool to plot the x1- and x2-values as functions of k. (There is no need to include this graph when you submit your answers.)
7. For what values of k is it best (in the sense of the largest value of A (x )) to spend more time studying math than chemistry? You must show your calculations.
8. Explain in words why this model predicts that it is best (again, in the sense of the largest value of A (x )) to spend the vast majority of your study time on chemistry when math is much easier than chemistry (k << 4) ANDE when math is much harder than chemistry (k >> 4).
Can anyone please help with this questions! Thanks so much =)
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