A manufacturer incurs production cost and transportation cost for manufacturing articles. The production cost, in dollars, for manufacturing n articles is given by the function f(n)=2n2−100n while transportation cost, in dollars, is given by the function g(n)=−n2−100n+20000. If the manufacturer sells each article for $100, which of the following represents the condition, in terms of n, for the manufacturer to make a profit? A n100 B n200 C 101≤n≤199 D 100

Question

A manufacturer incurs production cost and transportation cost for manufacturing articles. The production cost, in dollars, for manufacturing n articles is given by the function f(n)=2n2−100n  while transportation cost, in dollars, is given by the function g(n)=−n2−100n+20000. If the manufacturer sells each article for $100, which of the following represents the condition, in terms of n, for the manufacturer to make a profit?   A	n>100 B	n>200 C	101≤n≤199 D	100

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Determine Total Cost Function: Determine the total cost function by adding the production cost and transportation cost functions. Production cost function: f(n) = 2n^2 - 100n Transportation cost function: g(n) = -n^2 - 100n + 20000 Total cost function: h(n) = f(n) + g(n) h(n) = (2n^2 - 100n) + (-n^2 - 100n + 20000) h(n) = n^2 - 200n + 20000Determine Revenue Function: Determine the revenue function for selling n articles. Revenue function: R(n) = 100nSet Profit Inequality: Set up the inequality to find when the manufacturer makes a profit. Profit occurs when revenue is greater than total cost. R(n) > h(n) 100n > n^2 - 200n + 20000Rearrange Inequality: Rearrange the inequality to find the values of n for which the manufacturer makes a profit. 0 > n^2 - 300n + 20000 We need to solve for n in the quadratic inequality n^2 - 300n + 20000 < 0.Factor or Use Formula: Factor the quadratic inequality if possible or use the quadratic formula to find the roots. The quadratic factors as (n - 100)(n - 200) < 0.Determine True Intervals: Determine the intervals where the inequality holds true. The inequality (n - 100)(n - 200) < 0 is true when n is between the roots 100 and 200. So, the manufacturer makes a profit when n is greater than 100 and less than 200.Choose Correct Answer: Choose the correct answer from the given options that matches the interval found in Step 6. The correct answer is D, which states that the manufacturer makes a profit when 100 < n ≤ 200.

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