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If you order 10 or fewer of a certain textbook, they cost $50 each. However, if you order more than 10, then each textbook (starting with the 11th textbook) costs only $40 each, and if you order more than 20, then each textbook (starting with the 21st textbook) costs only $30 each.

Let n be the number of textbooks you buy.

a- Write down the total cost of the textbooks as a function of n.

a- Find the number of textbooks you must buy for the average cost of a textbook to be $42.

 May 16, 2023
 #1
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a) If you buy 10 or fewer textbooks, each textbook costs $50 each. If you buy more than 10, each textbook starting with the 11th textbook costs $40 each. And if you buy more than 20, each textbook starting with the 21st textbook costs $30 each.

The total cost of the textbooks as a function of n is therefore:

C(n) = 50n if n <= 10, 500 + 40(n - 10) if  10 < n <= 20, 1200 + 30(n - 20) if n > 20

b) The average cost of a textbook is $42 when the total cost of the textbooks is 42n.

If you buy 10 or fewer textbooks, the total cost is 50n, which is always greater than 42n. So the average cost of a textbook is never $42 if you buy 10 or fewer textbooks.

If you buy more than 10 but fewer than 20 textbooks, the total cost is 500+40(n−10). This is equal to 42n when n=15. So the average cost of a textbook is $42 when you buy 15 textbooks.

If you buy more than 20 textbooks, the total cost is 900+30(n−20). This is always less than 42n. So the average cost of a textbook is never $42 if you buy more than 20 textbooks.

Therefore, the number of textbooks you must buy for the average cost of a textbook to be $42 is n=15​.

 May 16, 2023
 #2
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Let f(n) be the total cost of n textbooks.

If n≤10, then each textbook costs $50, so f(n)=50n.

If 10

If n>20, then the first 10 textbooks cost $50 each, the next 10 textbooks cost $40 each, and the remaining n−20 textbooks cost $30 each, so f(n)=500+700+30(n−20).

To find the number of textbooks you must buy for the average cost of a textbook to be $42, we solve the equation f(n)/n=42.

If n≤10, then f(n)/n=50n/n=50>42, so we can rule out this case.

If 10 42 for 10

If n>20, then f(n)/n=(500+400+30(n−20))/n=900/n+30(n−20)/n. Since 900/n is a decreasing function of n and 30(n−20)/n is an increasing function of n, the graph of f(n)/n is a parabola that opens downwards. The vertex of the parabola is at n=30, where f(n)/n=40. Since f(n)/n<42 for n<30 and f(n)/n→30 as n→∞, there must be a unique value of n>30 for which f(n)/n=42. We can find this value of n numerically.

Solving the equation f(n)/n=42 for n>30, we find that n≈33.33. Therefore, you must buy 34​ textbooks for the average cost of a textbook to be $42.

 May 16, 2023

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