Question

The magic car company can sell a new car for

P(x) = 1500 - 3x... Fixed overhead = $60,000..... cost per car = $200.

a) Write down the formulas for R(x) and C(x). the revenue and cost functions, in terms of x, the number of cars produced and sold.

b) Let P(x) = R(x) - C(x) be the profit function. Sketch the graph of P(x) indicating the appropriate domain.

c) How many cars do you need to break even?

d) What is the maximum possible profit?

Answer 1

(a)

R(x) = x.P(x) = 1,500x - 3x²

C(x) = Fixed cost + Variable cost = 60,000 + 200x

(b)

P(x) = R(x) - C(x) = (1,500x - 3x²) - (60,000 + 200x) = 1,300x - 3x² - 60,000

When x = 0, P(x) = 0 (Vertical intercept of profit function curve).

Data table used:

x	P(x)
0	-60000
10	-47300
20	-35200
30	-23700
40	-12800
50	-2500
60	7200
70	16300
80	24800
90	32700
100	40000

Graph:

(c) In break-even, P(x) = 0.

1,300x - 3x² - 60,000 = 0

3x² - 1,300x + 60,000 = 0

Solving this quadratic equation using online solver,

x = 381 or x = 53 (considering integer values for quantity)

(d) Profit is maximized when dP(x)/dx = 0

1,300 - 6x = 0

6x = 1,300

x = 22 (considering integer values for quantity)

Profit = (1,300 x 22) - (3 x 22 x 22) - 60,000 = 28,600 - 1,452 - 60,000 = -32,852 (Loss)