Question

modeling a companys revenue and cost
how many units must the company sell to break even?
a company’s revenue, ( r(x) ), in thousands of dollars, from selling ( x ) thousand units is modeled by ( r(x) = -2x^2 + 12x ), and its cost, ( c(x) ), in thousands of dollars, is modeled by ( c(x) = 4x ).
2,000 or 4,000 units
0 or 4,000 units
0 or 16,000 units
2,000 or 3,000 units

Explanation:

Step1: Set Revenue = Cost

To break even, \( R(x) = C(x) \). Substitute \( R(x) = -2x^2 + 12x \) and \( C(x) = 4x \):
\( -2x^2 + 12x = 4x \)

Step2: Solve the Equation

Subtract \( 4x \) from both sides:
\( -2x^2 + 8x = 0 \)
Factor out \( -2x \):
\( -2x(x - 4) = 0 \)

Set each factor to zero:

1. \( -2x = 0 \Rightarrow x = 0 \)
2. \( x - 4 = 0 \Rightarrow x = 4 \)

Since \( x \) is in thousands of units, \( x = 0 \) means 0 units, and \( x = 4 \) means \( 4 \times 1000 = 4000 \) units.

Answer:

0 or 4,000 units