Question

Penguin Industries manufactures umbrellas.

The demand equation for umbrellas is given by p=20− x/100 for 0≤x≤1000 where p is the price in dollars and x is the number of umbrellas produced.
The cost of producing x umbrellas is given by C(x)=5000+5x.
(i) Show that the profit function is given by P(x)=15x−x²/100-5000.
(ii) Find the marginal profit when 200 umbrellas are produced and interpret the result.

Answer 1

The profit function is derived from the demand and cost functions, and the marginal profit can be found by taking the derivative of the profit function. Marginal profit represents the change in profit for each additional unit produced.

Step-by-step explanation:

To find the profit function, we need to subtract the cost function from the demand function. The demand equation is p = 20 - x/100, and the cost function is C(x) = 5000 + 5x.

(i) Profit function: P(x) = p(x) * x - C(x) = (20 - x/100) * x - (5000 + 5x) = 20x - (x²/100) - 5000.

(ii) To find the marginal profit, we take the derivative of the profit function with respect to x. P'(x) = 20 - (2x/100) = 20 - x/50. When 200 umbrellas are produced, the marginal profit is P'(200) = 20 - 200/50 = 16. This means that for each additional umbrella produced beyond 200, the profit decreases by $16.