Question

a product development manager is trying to determine the break - even point for a new service. the revenue function is $r(x)=-0.023x^{2}+240x$, and the cost function is $c(x)=58,000 + 98x$. in both functions, $x$ represents the number of units sold. according to the functions, what is the larger number of units of service the company can sell to break even? round to the nearest whole number if necessary. enter your answer in the box.

□ units

Explanation:

Step1: Set revenue equal to cost

$R(x) = C(x)$
$-0.023x^2 + 240x = 58000 + 98x$

Step2: Rearrange to standard quadratic form

Bring all terms to one side:
$-0.023x^2 + 240x - 98x - 58000 = 0$
$-0.023x^2 + 142x - 58000 = 0$
Multiply by -1 for simplicity:
$0.023x^2 - 142x + 58000 = 0$

Step3: Apply quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Here $a=0.023$, $b=-142$, $c=58000$:
$x=\frac{142\pm\sqrt{(-142)^2-4\times0.023\times58000}}{2\times0.023}$

Step4: Calculate discriminant

$\sqrt{142^2 - 4\times0.023\times58000}=\sqrt{20164 - 5336}=\sqrt{14828}\approx121.77$

Step5: Solve for both x values

First solution:
$x=\frac{142 + 121.77}{0.046}=\frac{263.77}{0.046}\approx5734$
Second solution:
$x=\frac{142 - 121.77}{0.046}=\frac{20.23}{0.046}\approx440$

Answer:

5734 units