Afertilizer producer finds that it can sell its product at a price of dollars per unit when it produces units of fertilizer. the total production cost (in dollars) for units is if the production capacity of the firm is at most 1000 units of fertilizer in a specified time, how many units must be manufactured and sold in that time to maximize the profit?

700

Explanation:

If from the question the price per unit of fertilizer, p(x) = 300 - 0.1x

The cost of x units of  fertilizer = C(x) = 15000 + 125x + 0.025x^2

And we know that Profit = Revenue - Cost

Revenue for x units of fertilizer R(X) = x*p(x)

= 300x - 0.1x^2

Hence Profit: P(x) = R(x) - C(x)

= [300x - 0.1x^2] - [15000 + 125x + 0.025x^2]  

= -0.125x^2 + 175x - 15000

To determine critical values

P'(x) = -0.25x + 175 = 0

Thus x = 175/0.25

x = 700

To maximize profit, the manufactures must produce 700 units of fertilizers

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