Question

a companys revenue, ( r(x) ), in thousands of dollars, from selling ( x ) thousand units is modeled by the quadratic function shown. ( r(x) = -x^2 + 15x - 28 ). its cost, ( c(x) ), in thousands of dollars, is modeled by the linear function ( c(x) = 4x ). how many units must the company sell to break even? 7,000 or 28,000 units 16,000 or 28,000 units 4,000 or 16,000 units 4,000 or 7,000 units

Explanation:

Step1: Set revenue equal to cost

Break-even occurs when $R(x) = C(x)$, so:
$-x^2 + 15x - 28 = 4x$

Step2: Rearrange into standard quadratic form

Subtract $4x$ from both sides to set equation to 0:
$-x^2 + 11x - 28 = 0$
Multiply by $-1$ to simplify:
$x^2 - 11x + 28 = 0$

Step3: Factor the quadratic equation

Find two numbers that multiply to 28 and add to -11:
$(x - 4)(x - 7) = 0$

Step4: Solve for x

Set each factor equal to 0:
$x - 4 = 0 \implies x = 4$
$x - 7 = 0 \implies x = 7$
Since $x$ is in thousands of units, multiply by 1000.

Answer:

4,000 or 7,000 units