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QuestionFor the function P(x)=−0.74x2+22x+75P(x)=-0.74 x^{2}+22 x+75, what constraints ensure profits are at least \175,000?Options:175,000? Options: -3.09 \leq x \leq 5.6or or 0 \leq x < 5.6$?

Studdy Solution

STEP 1

Assumptions1. The profit function is given by (x)=−0.74x+22x+75(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 xx for which the profit (x)(x) is at least 175175. Since the profits are given in thousands of dollars, we need to use 175175 (which represents 175,000175,000) in our calculations.
We set up the inequality (x)≥175(x) \geq175.
−0.74x2+22x+75≥175-0.74 x^{2}+22 x+75 \geq175

STEP 3

Next, we simplify the inequality by subtracting 175175 from both sides.
−0.74x2+22x+75−175≥0-0.74 x^{2}+22 x+75 -175 \geq0

STEP 4

implify the inequality.
−0.74x2+22x−100≥0-0.74 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 −0.74x2+22x−100=0-0.74 x^{2}+22 x -100 =0. We can use the quadratic formula for this, which is given byx=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}where a=−0.74a = -0.74, b=22b =22, and c=−100c = -100.

STEP 6

Substitute the values of aa, bb, and cc into the quadratic formula to find the roots.
x=−22±(22)2−4(−0.74)(−100)2(−0.74)x = \frac{-22 \pm \sqrt{(22)^{2}-4(-0.74)(-100)}}{2(-0.74)}

STEP 7

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

STEP 8

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

STEP 9

Calculate the two roots.
x=−22+188−.48x_{} = \frac{-22 + \sqrt{188}}{-.48}x2=−22−188−.48x_{2} = \frac{-22 - \sqrt{188}}{-.48}

STEP 10

Calculate the numerical values of the roots.
x=5.6x_{} =5.6x2=24.13x_{2} =24.13

STEP 11

The solutions to the inequality −0.74x+22x−100≥0-0.74 x^{}+22 x -100 \geq0 are the intervals (−∞,x]∪[x,∞)(-\infty, x_{}] \cup [x_{}, \infty), where xx_{} and xx_{} are the roots of the quadratic equation. However, since xx represents the number of calculators produced, it cannot be negative. Therefore, the reasonable constraints for the model are 0≤x≤5.60 \leq x \leq5.6 and 24.13≤x24.13 \leq x.
The correct answer is 0≤x≤5.60 \leq x \leq5.6 and 24.13≤x24.13 \leq x.

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