.Amanufacturercanproduceatmost60unitsofacertainproducteachyear.The demandequationfortheproductisp=q2 -50q+800andhisaveragefunctionis givenas: = -20q+ . ̅c 2 3 q2 5000 q a.Determinetheprofitmaximisingoutputandthecorrespondingmaximum profit. b.Ifthemanufacturerisnotrestrictedinhisproductioncapacity,how willit affectyouranswerin(a)above?

The question is asking for the profit maximizing output and the corresponding maximum profit for a manufacturer who can produce at most 60 units of a certain product each year. The demand equation for the product is p = q^2 - 50q + 800 and the average cost function is c̅ = -20q + (2/3)q^2 + 5000/q.

a. To find the profit maximizing output, we first need to find the revenue and cost functions.

The revenue (R) is price (p) times quantity (q), so R = pq. Substituting the demand equation into this gives R = q(q^2 - 50q + 800) = q^3 - 50q^2 + 800q.

The cost (C) is average cost (c̅) times quantity (q), so C = c̅q. Substituting the average cost function into this gives C = (-20q + (2/3)q^2 + 5000/q)q = -20q^2 + (2/3)q^3 + 5000.

The profit (P) is revenue (R) minus cost (C), so P = R - C = q^3 - 50q^2 + 800q - (-20q^2 + (2/3)q^3 + 5000) = (1/3)q^3 - 30q^2 + 800q - 5000.

To find the profit maximizing output, we take the derivative of the profit function and set it equal to zero: P' = q^2 - 60q + 800 = 0. Solving this quadratic equation gives q = 30 ± sqrt(100) = 30 ± 10. Since the manufacturer can produce at most 60 units, the profit maximizing output is q = 40 units.

Substituting q = 40 into the profit function gives P = (1/3)(40)^3 - 30(40)^2 + 800(40) - 5000 = $16,000.

b. If the manufacturer is not restricted in his production capacity, the profit maximizing output would be the same, q = 40 units, because this is the output level that maximizes profit. However, the maximum profit would be higher because the manufacturer could produce and sell more units.