Question

during a capital campaign, the business owners show investors information about a new products break - even point. the revenue function is $r(x)=-0.0075x^{2}+4,500x$. the cost includes a $210,000 fixed investment, along with $2,950 per unit produced. use a linear cost function where $x$ is the number of units sold. find the larger number of units the company can sell to break even. round your answer to the nearest thousand. enter your answer in the box. \boxed{} units

Explanation:

Step1: Define cost function

The linear cost function has fixed cost $210,000 and variable cost $2,950 per unit, so:
$C(x) = 2950x + 210000$

Step2: Set break-even condition

Break-even occurs when revenue equals cost: $R(x) = C(x)$
$-0.0075x^2 + 4500x = 2950x + 210000$

Step3: Rearrange to quadratic form

Bring all terms to one side to standard quadratic $ax^2+bx+c=0$:
$-0.0075x^2 + 4500x - 2950x - 210000 = 0$
$-0.0075x^2 + 1550x - 210000 = 0$
Multiply by -1 to simplify:
$0.0075x^2 - 1550x + 210000 = 0$

Step4: Apply quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=0.0075$, $b=-1550$, $c=210000$:
First calculate discriminant:
$\Delta = (-1550)^2 - 4\times0.0075\times210000$
$\Delta = 2402500 - 6300 = 2396200$
$\sqrt{\Delta} \approx 1547.966$
Now solve for x:
$x=\frac{1550\pm1547.966}{2\times0.0075}$

Step5: Calculate two solutions

First solution (using minus):
$x_1=\frac{1550 - 1547.966}{0.015} = \frac{2.034}{0.015} \approx 135.6$
Second solution (using plus):
$x_2=\frac{1550 + 1547.966}{0.015} = \frac{3097.966}{0.015} \approx 206531.07$

Step6: Round to nearest thousand

Round $206531.07$ to nearest thousand: 207000

Answer:

207000 units