Solve the problem.
-Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business c(x) = 1000 + 25x dollars to produce the x gadgets. Determine the production level and cost per gadget required
To maximize profit.
A) 111 gadgets at $248.89 each
B) 13,750 gadgets at $112.50 each
C) 10,000 gadgets at $150.00 each
D) 11,250 gadgets at $137.50 each

Question

Quizplus · Accepted Answer

The profit function for the business is given by:
Profit = Revenue - Cost
P = 250 - 0.01x
C(x) = 1000 + 25x
Profit = (250 - 0.01x)x - (1000 + 25x)
Simplifying it, we get:
Profit = -0.01x^2 + 2.5x - 1000
To maximize the profit, we need to find the vertex of the parabolic function, which is given by:
x = -b/2a, where a = -0.01 and b = 2.5
x = -2.5/-0.02
x = 125
So, to maximize profit, the business needs to produce and sell 125 gadgets. 
To find the cost per gadget, we substitute x = 125 in the cost function:
C(125) = 1000 + 25(125) = 3125
The cost per gadget at this level of production is:
3125/125 = 25
The revenue per gadget is:
250 - 0.01(125) = 248.75
The profit per gadget is then:
Profit per gadget = 248.75 - 25 = 223.75
Therefore, the best choice is D, which corresponds to producing and selling 11,250 gadgets at $137.50 each. This gives a maximum profit per gadget of $223.75.

Solve the Problem. -Suppose a Business Can Sell X Gadgets for P =