Question

modeling a restaurants revenue and cost
a restaurant’s revenue, $r(x) = -0.5x^2 + 5x$, in thousands of dollars, from selling $x$ thousand units, and its cost, $c(x) = 3x$, in thousands of dollars.
how many units must the restaurant sell to break even?
2,000 or 4,000 units
1,000 or 5,000 units
0 or 5,000 units
0 or 4,000 units

Explanation:

Step1: Set revenue equal to cost

$R(x) = C(x)$
$-0.5x^2 + 5x = 3x$

Step2: Rearrange into standard quadratic form

Subtract $3x$ from both sides:
$-0.5x^2 + 2x = 0$

Step3: Factor the quadratic equation

Factor out $-0.5x$:
$-0.5x(x - 4) = 0$

Step4: Solve for x

Set each factor equal to 0:
$-0.5x = 0 \implies x = 0$
$x - 4 = 0 \implies x = 4$
Since $x$ is in thousands of units, convert to actual units: $0 \times 1000 = 0$ units, $4 \times 1000 = 4000$ units.

Answer:

0 or 4,000 units