math help essay

Question

A manufacturing company wants to keep their revenue positive. The equation for C(x) represents their cost, where (x) represents the time in months. The equation for P(x) represents their profit. The equation for R(x) represents their revenue.

C(x) = 3x^(3) + 450x^(2)
R(x) = 7x^(3) + 300x^(2) + 900x
P(x) = R(x) - C(x)

a. Write an equation P(x) to represent the profit.
b. Identify the degree, leading coefficient, leading term, and constant of the profit equation.
c. Factor the polynomial.
d. Solve the equation to determine the values where the company will break even.

Answer 1

The equation for profit is:

 

P(x) = R(x) - C(x)    (profit = revenue - cost)

 

We have:

 

P(x) = 7x³ + 300x² + 900x - (3x³ + 450x²) = 4x³ - 150x² + 900x

 

This is a third degree polynomial, the leading coefficient is 7

 

There is no constant term. However P(x) can be factored as 2x(2x² -  75x + 450)

 

The company will break even when P(x) = 0

 

Obviously, the polynomial equals 0 when x = 0 (x represents the time in months)

 

Our focus is on 2x² - 75x + 450 = 0

 

We can use the Quadratic Formula to solve this. 

 

The roots are x = 30 and x = 15/2 = 7.5

 

The company will break even after 7.5 months, and after 30 months

 

 

We have already factored P(x) above