# maximum profit

If total cost, T(X), is related to output, X, by the equation:

T(X) = 0.001X2 + 0.5X + 50

a.What is the variable cost function?

b.What is the average cost function?

If output is 100 units, find

a.Total fixed cost;

b.Total variable cost;

c.Total cost;

d.Average cost;

e.Marginal cost of the 100th unit.

A firm can sell X units of a product at a price of P cents per unit, where P = 503 - X

The total costs of production is given by the function T(X) = 500 + 2X

a.Find the function that relates total revenue to the number of units sold - R(X).

b.Find the function that relates total profit to the number of units sold - P(X).

c.Find the number of units to be sold to achieve maximum profit and find the maximum profit.

d.What price per unit would the firm be charging at the maximum profit output level?

A Vancouver travel agency advertises all-expenses-paid trips to the Grey Cup Game for special groups. Transportation is by bus which seats 48 passengers, and the charge per person is $80, plus an additional $2 for each empty seat.

a.If there are X empty seats, find the equations for:

i.How many passengers are on the bus and

ii.How much does each pay?

b.What are the travel agency's total revenue when there are X empty seats?

c.Find the maximum revenue.

A quadratic revenue function often occurs in practice. The reason is that as more units are produced, the price must be lowered in order to sell them all.

A company has found that if they produce x units of a product, they must sell them each at $10 - 0.2x if they are to sell all of them.

a.Determine the revenue function.

b.Their fixed costs are $40 and their variable costs per unit are $1. Determine the cost and profit functions.

A small corner store owner decides to advertise on local radio in order to help stimulate his sales. Radio station CFAL charges him a flat rate of $99, plus $20 per day for advertising. From the onset of the advertising the revenue increase can be described as (120 - x) dollars per day where x is the number of days the ad is run. He plans to run the ad for only 60 consecutive days.

a.How many days must the ad run until the increased revenue just covers the cost of the advertising?

b.How many days should he let the ad run in order to maximize his profit from the ad campaign?

c.What is the maximum profit?

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#### Solution Preview

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Note: ^ means raised to the power of

For example X^2 means X square

Quadratic Functions

If total cost, T(X), is related to output, X, by the equation:

T(X) = 0.001X^2 + 0.5X + 50

a.What is the variable cost function?

Variable cost is that portion of the total cost function which does not have a constant term. This

VC(X) = 0.001X^2 + 0.5X

b.What is the average cost function?

Average Cost = Total Cost / X = 0.001X + 0.5 + 50/X

If output is 100 units, find

a.Total fixed cost;

Fixed portion of the total cost function

FC(X) =$50

b.Total variable cost;

VC(X) = 0.001X^2 + 0.5X = 0.001*100^2+0.5*100 = $60

c.Total cost;

T(X) = 0.001*100^2+0.5*100+50=$110

d.Average cost;

AC(X) = 0.001*100+0.5+50/100 = $1.10

e.Marginal cost of the 100th unit.

MC(X) = d[TC(X)]/dX = 0.001*2*X+0.5

MC(X) = 0.001*2*100+0.5= =$0.70

A firm can sell X units of a product at a price of P cents per unit, where P = 503 - X

The total costs of production is given by the function T(X) = 500 ...

#### Solution Summary

The maximum profit is discussed.

Managerial Economics: Calculating Maximum Profit and Revenue

A monopolist's demand function is given by

P = 80-3Q

(with MR = 80-6Q).

Its total cost function is

TC = 20Q + 200

(with MC = 20).

(i) Using algebra determine the profit maximizing output, price and optimal profit for the firm.

(ii) Suppose that instead of maximizing profit, the firm wants to maximize total revenue. Using algebra determine the optimal output, price, profit and revenue for the firm.

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