To determine the reasonable constraint for keeping production costs under $175,000, we set up the inequality
[tex]\[ C(x) < 175. \][/tex]
The function modeling the production cost is
[tex]\[ C(x) = -0.74x^2 + 22x + 75.\][/tex]
Let's break down the steps to find the reasonable constraint:
1. Set up the inequality:
[tex]\[-0.74x^2 + 22x + 75 < 175,\][/tex]
2. Move all terms to one side to form a quadratic inequality:
[tex]\[ -0.74x^2 + 22x + 75 - 175 < 0,\][/tex]
[tex]\[ -0.74x^2 + 22x - 100 < 0.\][/tex]
3. Solve the quadratic equation \( -0.74x^2 + 22x - 100 = 0 \) to find the critical points:
The quadratic equation can be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),
where \( a = -0.74 \), \( b = 22 \), and \( c = -100 \).
4. Calculate the discriminant:
[tex]\[\Delta = b^2 - 4ac,\][/tex]
[tex]\[ \Delta = 22^2 - 4(-0.74)(-100),\][/tex]
[tex]\[ \Delta = 484 - 296,\][/tex]
[tex]\[ \Delta = 188.\][/tex]
5. Calculate the roots:
[tex]\[ x = \frac{-22 \pm \sqrt{188}}{2(-0.74)},\][/tex]
[tex]\[ x = \frac{-22 \pm \sqrt{188}}{-1.48}.\][/tex]
Therefore, the roots will be:
[tex]\[
x_1 = \frac{-22 + \sqrt{188}}{-1.48}, \quad x_2 = \frac{-22 - \sqrt{188}}{-1.48}.
\][/tex]
6. Simplify the solutions:
Using the approximations for the square roots and simplifying, we get:
[tex]\[ x_1 \approx 5.6,\][/tex]
[tex]\[ x_2 \approx 24.13.\][/tex]
7. Interpret the inequality:
The quadratic inequality \(-0.74x^2 + 22x - 100 < 0\) is satisfied for the values of \(x\) between the roots where the parabola opens downwards:
[tex]\[ 5.6 < x < 24.13. \][/tex]
Therefore, the intervals for which the production costs are under $175,000 are:
[tex]\[ 5.6 < x < 24.13. \][/tex]
Therefore, the reasonable constraint for the model keeping production costs under $175,000 is:
[tex]\[ 5.6 < x < 24.13. \][/tex]
