Question

a small - business owner is trying to determine the break - even point. the revenue function is $r(x)=-0.125x^{2}+250x$, where $x$ is the number of units produced. the table lists the cost based on the number of units produced.

units	cost ($)
1,250	67,900

use the table to write the linear cost function in terms of $x$, the number of units produced. what is the larger number of units the owner can sell to break even? round to the nearest whole number.
73,432 units
3 units
1,566 units
784 units

Explanation:

Step1: Define linear cost function

A linear cost function has the form $C(x) = mx + b$, where $m$ is the marginal cost per unit, and $b$ is fixed costs.

Step2: Calculate slope $m$

Use the two points $(500, 27400)$ and $(1250, 67900)$:
$$m = \frac{67900 - 27400}{1250 - 500} = \frac{40500}{750} = 54$$

Step3: Solve for fixed cost $b$

Substitute $x=500$, $C(x)=27400$, $m=54$ into $C(x)=mx+b$:
$$27400 = 54(500) + b$$
$$27400 = 27000 + b$$
$$b = 27400 - 27000 = 400$$
So $C(x) = 54x + 400$.

Step4: Set revenue = cost (break-even)

$$-0.125x^2 + 250x = 54x + 400$$
Rearrange into standard quadratic form:
$$-0.125x^2 + 196x - 400 = 0$$
Multiply through by $-8$ to eliminate decimals:
$$x^2 - 1568x + 3200 = 0$$

Step5: Solve quadratic equation

Use quadratic formula $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$, where $A=1$, $B=-1568$, $C=3200$:
First calculate discriminant:
$$\Delta = (-1568)^2 - 4(1)(3200) = 2458624 - 12800 = 2445824$$
$$\sqrt{\Delta} \approx 1563.91$$
Find the two solutions:
$$x_1 = \frac{1568 + 1563.91}{2} \approx \frac{3131.91}{2} \approx 1566$$
$$x_2 = \frac{1568 - 1563.91}{2} \approx \frac{4.09}{2} \approx 2$$

Step6: Identify larger solution

The larger $x$ value is $\approx 1566$.

Answer:

1,566 units