Suppose a business can sell x gadgets for p(x)=250 - 0.01x dollars a piece, and it costs the business c(x) = 1,000 + 25x dollars so produce the x gadgets. Determine the production level and cost per gadget required to maximize profit.

A. 13,750 gadget at $112.50 each
B. 111 gadgets at $248.89 each
C. 11, 250 gadgets at $137.50 each
D. 10,000 gadgets at $150.00 each

A

Explanation:

In this question, we are asked to determine the production level and cost per gadget required for profit maximization.

p(x) = 250 - 0.01x

c(x) = 1000+25x

The total amount made from selling a number of x gadgets= number of gadgets * cost at which the gadget is sold per piece

Mathematically = r(x) = x p(x) = 250x - 0.01x^2

The profit made is equate to the revenue minus the cost of production

Mathematically

p(x) = r(x) - c(x) = 250x - 0.01x^2 - (1000+25x) = 225x - 0.01x^2 -1000

To get maximum profit, the first derivative of the profit must be equal to 0

p'(x) = 225 - 0.02x

p'(x)=0 gives 225-0.02x=0

=> x = 11250

and p(x) = 250 - 0.01*11250 = 137.50

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