Answer:
The number of boxes of washers Washer King should produce per day to maximize profit = 960 boxes.
And the corresponding maximum daily profit = $2,114
Explanation:
The daily, short-run total cost of producing Q boxes of the product is given as
TC = 190 + 0.20Q + 0.0025Q²
The unit price of the product = $5.
Total revenue = (Unit Price) × (Quantity sold) = 5Q
Profit = (Revenue) - (Total Cost)
Profit = 5Q - (190 + 0.20Q + 0.0025Q²)
Profit = P(Q) = -190 + 4.8Q - 0.0025Q²
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) = -190 + 4.8Q - 0.0025Q²
(dP/dQ) = 4.8 - 0.005Q
At maximum point,
(dP/dQ) = 4.8 - 0.005Q = 0
Q = (4.8/0.005) = 960 boxes
(d²P/dQ²) = -0.005 < 0 (hence, showing that the this point corresponds to a maximum point truly)
Hence, the number of boxes of washers Washer King should produce per day to maximize profit = 960 boxes.
The corresponding maximum profit is then obtained from
P(960) = -190 + (4.8×960) - 0.0025(960²)
Maximum daily profit = $2,114
Hope this Helps!!!
