If a company charges x dollars per item, it finds that it can sell 1500 - 3x of them. each item costs $8 to produce.
(a) express the revenue, r(x), as the function of price.
(b) express the cost, c(x), as a function of price.
(c) express the profit, p(x), which is revenue minus cost, as a function of price.

The profit is $496.30

a. The revenue function will be calculated thus:

R(x) = (1500 - 3x) × x

R(x) = 1500x - 3x²

b. The cost function will be:

C(x) = 8 × x = 8x

c. The profit function will be:

P(x) = Revenue - Cost

= 1500x - 3x² - (8x)

= 1500x - 3x² - 8x

Divide through by x

= (1500x - 3x² - 8x) / x

= 1500 - 3x - 8

1500 - 3x - 8 = 0

Collect like terms

3x = 1500 - 8.

3x = 1492

x = 1492/3

x = 496.3

The profit is $496.30

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Step-by-step explanation:

According to that the number of produced items is (1500-3x) where x is the price

(a) the revenue is price*production

(1500-3x)x = 1500x-3x^2

(b) the cost is unitary cost*production

8x

(c) the profit is revenue - cost

1500x-3x^2 - 8x = 1492x - 3x^2

And according to that maximum profit is reached when the price is $249