Answer:
Profit at the optimal integer output level is $176.50.
Explanation:
This can be determined as follows:
Step 1: Calculation of optimal integer output level
At the optimal integer output level, profit is maximized where marginal revenue (MR) is equal to the marginal cost (MC), i.e. where;
MR = MC ................................ (1)
For any product, the MR is equal to the price per unit of the product. Therefore, we have:
MR = Price per unit = $60
Also given,
MC = 7q
Substituting for MR and MC in equation (1) and solve for q, we have:
$60 = 7q
q = $60 / 7
q = 9 units
Therefore, the optimal integer output level is 9 units.
Step 2: Calculation of total revenue at optimal integer output level
Total revenue = Price per unit * q = $60 * 9 = $540
Step 3: Calculation of total cost at optimal integer output level
Since MC = 7q, the total cost (C) can be obtained by taking the integral of the MC as follows:
C = ∫(MC)dq = ∫[7q]dq = (7 / 2)q^2 + F = 3.5q^2 + F ........... (2)
Where F is Fixed cost which is given as $80.
We then substitute F = $80 and q = 9 into equation (2) to have:
C = 3.5(9^2) + 80
C = (3.5 * 81) + 80
C = 283.50 + 80
C = $363.50
Therefore, total cost at the optimal integer output level is $363.50.
Step 4: Calculation of profit at optimal integer output level
Profit = Total revenue - Total cost ...................... (3)
Where;
Total revenue = $540; from step 2 above.
Total cost = $363.50; form step 3 above.
Substituting the values into equation (3), we have:
Profit = $540 - $363.50 = $176.50
Therefore, profit at the optimal integer output level is $176.50.
