Business Math Word Problem

Question

a company produces 10 items, charges $45 per item. when it produces 20 items, it charges $40 per item. cost to produce 10 items is $450, cost to produce 20 items is $650. what is the price, cost, revenue, profit functions. what is the break-even points, max profit and items to produce to get max profit.

 

 

Answer 1

Price Slope = (change in Price)/(change in quantity) = (40 - 45)/(20 - 10) = -5/10 = -.5

 

y-intercept = 45 + .5(10) = 50

 

P is for price 

 

Q is for quantity

 

Price equation: P = $50 - $0.50Q

 

Cost Slope = (change in cost)/(change in quantity) = (650 - 450)/(20 -10) = 200/10 = 20

 

y-intercept = 450 - 20(10) = 250

 

Cost equation = $250 + $20Q

 

Revenue equation = Price * Quantity = ($50 - $0.50Q) * Q = 50Q - 0.50Q^2

 

Profit equation = Revenue - Cost = (50Q - 0.50Q^2) - (250 + 20Q) = -250 + 30Q - 0.50Q^2

 

Break even point is when revenue equals cost of a certain item

 

50Q - 50Q^2 = 250 + 20Q

 

.50Q^2 - 30Q + 250= 0

 

Q^2 - 60Q + 500 = 0

 

(Q - 50)(Q - 10) = 0

 

Q = 50 , Q = 10

 

When quantity is 10 or 50 items, there is no profit or loss. It breaks even. 

 

Maximized profit

 

If we use calculus, then we can take a derivative of the profit function to see where it is maximized.

 

Profit = -250 + 30Q - 0.5Q^2

 

P' = 30 - Q

 

30 - Q = 0

 

Q = 30

 

The slope goes from positive to negative when it passes through Q=30 which denotes a maximum. Therefore the maximum profit occurs at a quantity of 30. The maximum profit is:

 

-250 + 30(30) - .50(30)^2 = -250 + 900 - 450 = $200.