Math Snap

STEP 1

Assumptions1. The profit function is given by $(x)=-0.74 x^{}+22 x+75$
. The profits, $$, are in thousands of dollars3. $x$ is the number of calculators produced, in thousands4. The company wants to keep its profits at or above $\($\)175,000$

STEP 2

First, we need to find the values of $x$ for which the profit $(x)$ is at least $175$. Since the profits are given in thousands of dollars, we need to use $175$ (which represents $175,000$) in our calculations.
We set up the inequality $(x) \geq175$.
$$-0.74 x^{2}+22 x+75 \geq175$$

STEP 3

Next, we simplify the inequality by subtracting $175$ from both sides.
$$-0.74 x^{2}+22 x+75 -175 \geq0$$

STEP 4

implify the inequality.
$$-0.74 x^{2}+22 x -100 \geq0$$

STEP 5

Now we need to solve this quadratic inequality. To do this, we first find the roots of the corresponding quadratic equation $-0.74 x^{2}+22 x -100 =0$. We can use the quadratic formula for this, which is given by$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$where $a = -0.74$, $b =22$, and $c = -100$.

STEP 6

Substitute the values of $a$, $b$, and $c$ into the quadratic formula to find the roots.
$$x = \frac{-22 \pm \sqrt{(22)^{2}-4(-0.74)(-100)}}{2(-0.74)}$$

STEP 7

implify the expression under the square root.
$$x = \frac{-22 \pm \sqrt{484-296}}{-1.48}$$

STEP 8

Further simplify the expression.
$$x = \frac{-22 \pm \sqrt{188}}{-1.48}$$

STEP 9

Calculate the two roots.
$$x_{} = \frac{-22 + \sqrt{188}}{-.48}$$$$x_{2} = \frac{-22 - \sqrt{188}}{-.48}$$

STEP 10

Calculate the numerical values of the roots.
$$x_{} =5.6$$$$x_{2} =24.13$$

The solutions to the inequality $-0.74 x^{}+22 x -100 \geq0$ are the intervals $(-\infty, x_{}] \cup [x_{}, \infty)$, where $x_{}$ and $x_{}$ are the roots of the quadratic equation. However, since $x$ represents the number of calculators produced, it cannot be negative. Therefore, the reasonable constraints for the model are $0 \leq x \leq5.6$ and $24.13 \leq x$.
The correct answer is $0 \leq x \leq5.6$ and $24.13 \leq x$.