Question

For the cost and price functions below, find

Answer 1

Answer:

a) The number, q, of units that produces maximum profit = 15

b) The price, p, per unit that produces maximum profit = 47 (currency not giben in the question)

c) Maximum Profit = P = 380 (currency not given in the question).

Explanation:

The cost function and price per unit function are given respectively as

C(q) = 70 + 17q

p = 77 - 2q

where q = quantity or number of units

a.) the number, q, of units that produces maximum profit

Total cost = C(q) = 70 + 17q

Revenue = (price per unit) × (Number of units) = p × q = (77 - 2q) × q = (77q - 2q²)

Profits = P(q) = (Revenue) - (Total Cost)

P(q) = (77q - 2q²) - (70 + 17q)

P(q) = -2q² + 60q - 70

To maximize the profits, we just obtain the point where the profit function reaches a Maximum.

At the maximum of a function, (dP/dq) = 0 and (d²P/dq²) < 0

Profit = P(q) = -2q² + 60q - 70

(dP/dq) = -4q + 60

At maximum point,

(dP/dq) = -4q + 60 = 0

q = (60/4) = 15

(d²P/dQ²) = -4 < 0 (hence, showing that the this point corresponds to a maximum point truly)

Hence, the number, q, of units that produces maximum profit = 15.

b.) the price, p, per unit that produces maximum profit

The price per unit is given as

p = 77 - 2q

Maximum profit occurs at q = 15

p = 77 - (2×15) = 47

Hence, the price, p, per unit that produces maximum profit = 47 (currency not given in the question)

c.) the maximum profit, P.

The Profit function is given as

Profit = P(q) = -2q² + 60q - 70

At maximum Profit, q = 15

Maximum Profit = P(15)

= -2(15²) + 60(15) - 70

= 380 (currency not given in the question).

Hope this Helps!!!