A company sells running shoes to dealers at a rate of $36 per pair if fewer than 75 pairs are ordered. If a de?

A company sells running shoes to dealers at a rate of $36 per pair if fewer than 75 pairs are ordered. If a dealer orders 75 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company? Please show steps!!
Answer:

Let n be the number of pairs of running shoes ordered. For the following calculation to be valid 75 ≤ n ≤ 600. The price per pair = 36 - 0.04n Then monies received , M = number ordered x price. M = n(36 - 0.04n) = 36n - 0.04n² M is a maximum (or minimum) when dM/dn = 0, that is when :- 36 - 0.08n = 0 : 0.08n = 36 : n = 450. [Note : as d²M/dn² = -0.08 and this is always negative then the situation when dM/dn = 0 is a MAXIMUM] Then an order for 450 pairs will produce the maximum income. The money generated will be 450 [36 - (0.04 x 450)] = 450[36 - 18] = 450 x 18 = $8100.

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Let n be the number of pairs of running shoes ordered. For the following calculation to be valid 75 ≤ n ≤ 600. The price per pair = 36 - 0.04n Then monies received , M = number ordered x price. M = n(36 - 0.04n) = 36n - 0.04n² M is a maximum (or minimum) when dM/dn = 0, that is when :- 36 - 0.08n = 0 : 0.08n = 36 : n = 450. [Note : as d²M/dn² = -0.08 and this is always negative then the situation when dM/dn = 0 is a MAXIMUM] Then an order for 450 pairs will produce the maximum income. The money generated will be 450 [36 - (0.04 x 450)] = 450[36 - 18] = 450 x 18 = $8100.

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