49) the highest selling price at which x units of manufacturer\'s prodcut can be

Question

49) the highest selling price at which x units of manufacturer's prodcut can be sold is p(x)=240-0.01x dollars per unit and the cost of producing x units of this product is C(x) = 40x+8000 dollars.

a. determine the manufactuer's profit function, p(x)

b. determine how many units of this product need to be produced and sold in order to earn the maximum profit.

c. what selling price will produce this maximum profit?

d. what is the maximum profit

Solutions:

a. P(x)=-0.01x^2+200x-8000

b. 10,000 units

c. $140 per unit

d. $992,000

Explanation / Answer

Solution:

Given; p(x)=240-0.01x and C(x) = 40x+8000.

(a)

Profit function P(x) = R - C = xp - C
P(x) = x * (240-0.01x) - (40x + 8000)
P(x) = 240x - 0.01x^2 - 40x - 8000
P(x) = -0.01x^2 + 200x - 8000

(b)

This has a maximum when its derivative is zero:

P'(x) = -0.02x + 200 = 0
=> 0.02x = 200
=> x = 200/0.02 = 10,000 units

(c)

selling price will produce this maximum profit;

p(10000) = 240 - 0.01*10000 = $140 per unit

(d)

maximum profit at x = 10,000 units

P(x)=-0.01x^2+200x-8000

P(x) = -0.01(10000)^2 + 200*10000 - 8000

P(x) = -1000000 + 2000000 - 8000

P(x) = $ 992000

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