A manufacturer finds it costs him x + 5x + 7 dollars to produce x tons of an item. At 2 production levels above 3 tons, he must hire additional workers, and his costs increase by 3 (x - 3) dollars on his total production. If the price he receives is $13 per ton regardless of how much he manufactures and if he has a plant capacity of 10 tons, what level of output maximizes his profits?

Question

Mathematics

Annabelle

21 October, 07:52

A manufacturer finds it costs him x + 5x + 7 dollars to produce x tons of an item. At 2 production levels above 3 tons, he must hire additional workers, and his costs increase by 3 (x - 3) dollars on his total production. If the price he receives is $13 per ton regardless of how much he manufactures and if he has a plant capacity of 10 tons, what level of output maximizes his profits?

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Answers (1)

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Luna Sexton · Answer 1

Answer: The maximum = 3 tons

Step-by-step explanation:

The cost function C (x) = x + 5x + 7

P (*) = 13x - x^2 - 5x - 7

If x <3

P = x^2 + 8x - 7

Differentiating to get x

dp/dx = - 2x + 8

X = 8/2

C=4

Maximum will be 3 tons

When x=3

P = 13x - x^2 - 5x + 7 - 3x + 9

When x>3

dp/dx = x^2+5x + 2

X = 5/2 = 2.5