Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
[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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
