Question

a business owner is trying to determine the break - even point for a new luxury product offered in limited locations. the revenue function is $r(x)= - 0.009x^{2}+12,000x$. the cost includes a fixed investment of $760,000, along with $9,750 per unit. write a linear cost function and use it to find the larger number of units the company can sell to break even. round to the nearest thousand. enter your answer in the box. \boxed{} units

Explanation:

Step1: Define linear cost function

Cost = Fixed cost + Variable cost. Fixed cost is $760,000, variable cost is $9,750 per unit.
$C(x) = 9750x + 760000$

Step2: Set break-even condition

Break-even occurs when $R(x)=C(x)$.
$-0.009x^2 + 12000x = 9750x + 760000$

Step3: Rearrange to standard quadratic form

Bring all terms to one side to get $ax^2+bx+c=0$.
$-0.009x^2 + 12000x - 9750x - 760000 = 0$
$-0.009x^2 + 2250x - 760000 = 0$
Multiply by -1: $0.009x^2 - 2250x + 760000 = 0$

Step4: Apply quadratic formula

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a=0.009$, $b=-2250$, $c=760000$.
First calculate discriminant:
$\Delta = (-2250)^2 - 4(0.009)(760000) = 5062500 - 27360 = 5035140$
Then solve for x:
$x = \frac{2250 \pm \sqrt{5035140}}{2(0.009)}$
$\sqrt{5035140} \approx 2243.91$

Step5: Calculate two solutions

First solution (using minus):
$x_1 = \frac{2250 - 2243.91}{0.018} \approx \frac{6.09}{0.018} \approx 338.33$
Second solution (using plus):
$x_2 = \frac{2250 + 2243.91}{0.018} \approx \frac{4493.91}{0.018} \approx 249661.67$

Step6: Round to nearest thousand

Round $249661.67$ to nearest thousand: $250000$

Answer:

250000 units