When does a company get the maximum profit if: Marginal Cost = dC/dq = 6q + 20 and P = 100-q, where q = output. Fixed costs are 600.
From the information provided I assume you want the number of units produced to maximise profit.
The marginal cost is given as dC/dq = 6q + 20
Total cost is the integral of the marginal cost or C = Int[6q + 20] = 6q^2/2 + 20q + F = 3q^2 + 20q + F
As the fixed costs are 600, F = 600 in the function obtained earlier.
The total cost for q units is 3q^2 + 20q + 600.
I assume the revenue per unit is 100 - q. The revenue when q units are produced 100q - q^2.
Total profit for q units is TP = Revenue - Total cost
=> 100q - q^2 - (3q^2 + 20q + 600)
=> 100q - q^2 - 3q^2 - 20q - 600
=> -4q^2 + 80q - 600
For profit maximization we need to differentiate -4q^2 + 80q - 600 with respect to q and solve for q.
d (TP) / dq = -8q + 80
-8q + 80 = 0
=> q = 10
The company makes the maximum profit when it manufactures 10 units.