Question

a companys revenue, r(x), in thousands of dollars, from selling x thousand units is modeled by the quadratic function shown. r(x) = -x² + 12x - 18. its cost, c(x), in thousands of dollars, is modeled by the linear function c(x) = 3x. how many units must the company sell to break even? 3,000 or 6,000 units 3,000 or 9,000 units 9,000 or 18,000 units 0 or 6,000 units

Explanation:

Step1: Set revenue equal to cost

$R(x) = C(x)$
$-x^2 + 12x - 18 = 3x$

Step2: Rearrange to standard quadratic form

$-x^2 + 9x - 18 = 0$
Multiply by $-1$: $x^2 - 9x + 18 = 0$

Step3: Factor the quadratic equation

$(x - 3)(x - 6) = 0$

Step4: Solve for x

$x - 3 = 0 \implies x=3$; $x - 6 = 0 \implies x=6$

Step5: Convert to actual units

Since $x$ is in thousands of units, $3 \times 1000 = 3000$ and $6 \times 1000 = 6000$

Answer:

3,000 or 6,000 units