Suppose we have the following Cost and Revenue Functions. How might we create a Profit Function P(x)?

Question

Suppose we have the following Cost and Revenue Functions:

C(x) = 2x2 + 90 and R(x) = x2 + 20x

How might we create a Profit Function P(x)? What would it be?

Interpret P(8)=?

P(11)=?

P(13)=?

P(15)=?

Answer 1

Hi,

A profit function P(x) would be the Revenue Function minus the Cost Function. This would be:

P(x) = R(x) - C(x) =
x² + 20x - (2x² + 90) =
-x² + 20x - 90

So P(x) = -x² + 20x - 90

P(8) = -8² + 20(8) - 90 = -64 + 160 - 90 = 6 profit

P(11) = -11² + 20(11) - 90 = -121 + 220 - 90 = 9 profit

P(13) = -13² + 20(13) - 90 = -169 + 260 - 90 = 1 profit

P(15) = -15² + 20(15) - 90 = -225 + 300 - 90 = -15 loss

I hope that helps!! :-)

Answer 2

P(x)=R(x)-C(x)=2x^2+90-x^2-20x
x2=x^2 ?
P(x)=x^2-20x+90
p(11)=11^2-20(11)+90
121-220+90=189
P(13) and P(15) are similar.

Answer 3

P(x) = R(x) - C(x)
P(x) = x^2+20x - 2x^2 -90
P(x) = -x^2 +20x - 90
P(8) = 6
P(11) = 9
P(13) = 1
P(15) = -15
Max profit occurs when x = 10 and equals 10

Answer 4

P(x) = x2 + 20x - (2x2 + 90)
P(x) = -x2 + 20x - 90

dP/dx = -2x + 20
let dP/dx = 0
2x=20
x=10 (max profit point)
max profit = -100 + 200 = 100

Answer 5

P(x) = R(x) - C(x)