Question

(a) The total cost, in millions of dollars, of producing x thousand units of an item is C(x) = 4(x - 5)2 + 4. Plot at least 2 points on the function C(x) with x on the horizontal axis. Then click Connect Points. Plot the vertex and 2 additional points. Vertex Point Connect Points Delete Reset Vertex 2 9.3, 20.9

Answer 1

a. C(x) = 4(x-5)2+4 is the given cost function

If x=4, C = 4(4-5)2+4 = 8

If x=6, C = 4(6-5)2+4 = 8

If x=2, C = 4(2-5)2+4 = 40

If x=8, C = 4(8-5)2+4 = 40

If x=5, C = 4(5-5)2+4 = 4

Plotting the points (2,40),(4,8), (5,4), (6,8) and (8,40), we get U-shaped cost function.

b. R(x) = 5x is the given revenue function.

Now, revenue = price * quantity . This means price = $5 per unit (constant).

c. Profit = R(x) - C(x)

or, Profit = 5x-{4(x-5)2+4}

or, Profit = 5x-(4x2-40x+100+4)

or, Profit = 5x-4x2+40x-104

or, Profit = -4x2+45x-104

i) Now, to earn profit, profit > 0

or, -4x2+45x-104 > 0

or, 4x2-45x+104 < 0

or, (4x-13) (x-8) < 0

or, (x-3.25) (x-8) < 0

Now, profit will be positive when 3.25<x<8

ii) Similarly, the firm will break even if profit = 0

or, -4x2+45x-104 = 0

or, 4x2-45x+104 = 0

or, (4x-13) (x-8) = 0

or, (x-3.25) (x-8) = 0

Now, firm will break-even if x=3.25 and if x=8

iii) Again, the firm will loose money if profit < 0

or, -4x2+45x-104 < 0

or, 4x2-45x+104 > 0

or, (4x-13) (x-8) > 0

or, (x-3.25) (x-8) > 0

Now, profit will be positive when x<3.25 and x>8