The graph of the function [tex][tex]$C(x) = -0.52x^2 + 23x + 92$[/tex][/tex] is shown. The function models the production cost, [tex][tex]$C$[/tex][/tex], in thousands of dollars for a candle company to manufacture a candle, where [tex][tex]$x$[/tex][/tex] is the number of candles produced, in thousands.

If the company wants to keep its production costs under [tex]$225,000, then which constraint is reasonable for the model?

A. [tex]$[/tex]0 \leq x < 6.84[tex]$[/tex] and [tex]$[/tex]37.39 < x \leq 47.92[tex]$[/tex]
B. [tex]$[/tex]-3.69 \leq x \leq 6.84[tex]$[/tex] and [tex]$[/tex]37.30 < x \leq 47.92[tex]$[/tex]
C. [tex]$[/tex]-3.69 \leq x \leq 47.92$[/tex]



Answer :

To find the production levels that keep the production costs under [tex]$225,000$[/tex], we need to solve the inequality:

[tex]\[ C(x) = -0.52x^2 + 23x + 92 < 225 \][/tex]

First, we rearrange this inequality into a standard quadratic form:

[tex]\[ -0.52x^2 + 23x + 92 - 225 < 0 \][/tex]

Simplify the constant term:

[tex]\[ -0.52x^2 + 23x - 133 < 0 \][/tex]

Next, we need to find the roots of the quadratic equation:

[tex]\[ -0.52x^2 + 23x - 133 = 0 \][/tex]

We can use the quadratic formula, which is:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For our quadratic equation [tex]\( -0.52x^2 + 23x - 133 = 0 \)[/tex], the coefficients are:
- [tex]\(a = -0.52\)[/tex]
- [tex]\(b = 23\)[/tex]
- [tex]\(c = -133\)[/tex]

Using the quadratic formula, we find:

[tex]\[ x = \frac{-23 \pm \sqrt{23^2 - 4(-0.52)(-133)}}{2(-0.52)} \][/tex]

Calculate the discriminant:

[tex]\[ \Delta = 23^2 - 4 \cdot (-0.52) \cdot (-133) \][/tex]
[tex]\[ \Delta = 529 - 2.08 \cdot 133 \][/tex]
[tex]\[ \Delta = 529 - 276.64 \][/tex]
[tex]\[ \Delta = 252.36 \][/tex]

Now plug this back into the quadratic formula:

[tex]\[ x = \frac{-23 \pm \sqrt{252.36}}{-1.04} \][/tex]
[tex]\[ x = \frac{-23 \pm 15.88}{-1.04} \][/tex]

This gives us two solutions:

[tex]\[ x_1 = \frac{-23 + 15.88}{-1.04} \][/tex]
[tex]\[ x_1 = \frac{-7.12}{-1.04} \][/tex]
[tex]\[ x_1 \approx 6.84 \][/tex]

[tex]\[ x_2 = \frac{-23 - 15.88}{-1.04} \][/tex]
[tex]\[ x_2 = \frac{-38.88}{-1.04} \][/tex]
[tex]\[ x_2 \approx 37.38 \][/tex]

The roots of the quadratic equation are approximately [tex]\( x = 6.84 \)[/tex] and [tex]\( x = 37.38 \)[/tex].

Since the quadratic equation opens downward (the coefficient of [tex]\( x^2 \)[/tex] is negative), the inequality [tex]\( -0.52x^2 + 23x - 133 < 0 \)[/tex] is satisfied between the roots. Therefore,

[tex]\[ 6.84 < x < 37.38 \][/tex]

So, the constraint that keeps production costs under \$225,000 is [tex]\( 6.84 < x < 37.38 \)[/tex].

None of the options exactly match this solution, but the closest in terms of the valid intervals given would be:

[tex]\[ 0 \leq x < 6.84 \quad \text{and} \quad 37.39 < x \leq 47.92 \][/tex]

This option includes intervals that exclude the costly production between [tex]\( 6.84 \)[/tex] and [tex]\( 37.38 \)[/tex].