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.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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