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!!!
