Question
The production cost, C, in thousands of dollars, for a food company to make bags of potato chips is given by the model C(x) = −0.16x2 + 6.4x + 36, where x is the number of bags of chips produced in one day, in thousands. The company wants to keep its production cost below $30,000. The graph shown models the situation.

Constraints are restrictions that need to be placed upon variables used in equations. Maximum and minimum values, domain and range, and asymptotes are all examples of constraints placed on a function. What are all the reasonable constraints for the model?
Select TWO correct answers.
A.
x < −0.92
B.
x > −0.92
C.
x > 40.92
D.
x < 40.92
E.
x ≤ 45

Constraints are restrictions that need to be placed upon variables used in equations. Maximum and minimum values, domain and range, and asymptotes are all examples of constraints placed on a function. What are all the reasonable constraints for the model?
Select TWO correct answers.
A.
x < −0.92
B.
x > −0.92
C.
x > 40.92
D.
x < 40.92
E.
x ≤ 45
Answers
Answered by
GPT-4o mini
To analyze the constraints for the production cost model \( C(x) = -0.16x^2 + 6.4x + 36 \), we first need to understand the context of the variable \( x \). Here, \( x \) represents the number of bags of chips produced in thousands, which means \( x \) should be a non-negative number since production cannot be negative.
Additionally, the company wishes to keep the production cost below $30,000. Thus, we set up the inequality:
\[
C(x) < 30
\]
Substituting the cost function into the inequality gives us:
\[
-0.16x^2 + 6.4x + 36 < 30
\]
Simplifying this:
\[
-0.16x^2 + 6.4x + 6 < 0
\]
To solve for \( x \), we can rearrange the equation:
\[
-0.16x^2 + 6.4x + 6 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.16 \), \( b = 6.4 \), and \( c = 6 \), we find the roots.
Calculating the discriminant:
\[
D = b^2 - 4ac = (6.4)^2 - 4(-0.16)(6) = 40.96 + 9.6 = 50.56
\]
Finding the roots:
\[
x = \frac{-6.4 \pm \sqrt{50.56}}{2(-0.16)}
\]
Calculating:
\[
\sqrt{50.56} \approx 7.1
\]
This leads to two roots:
\[
x = \frac{-6.4 + 7.1}{-0.32} \text{and} x = \frac{-6.4 - 7.1}{-0.32}
\]
Calculating the first root:
\[
x_1 \approx \frac{0.7}{-0.32} \approx -2.19
\]
Calculating the second root:
\[
x_2 \approx \frac{-13.5}{-0.32} \approx 42.19
\]
Evaluating the inequality \( C(x) < 30 \), we find that \( x \) must be in the range:
\[
-2.19 < x < 42.19
\]
Since \( x \) has to be greater than or equal to zero, we further restrict our domain to:
\[
0 \leq x < 42.19
\]
The reasonable constraints for the model, therefore, are:
1. \( x \geq 0 \)
2. From the roots, we have \( x < 42.19 \), or rounded, \( x \leq 42 \) or similar.
Based on the options presented, the correct constraints from the given options should be:
**D. \( x < 40.92 \)** (since it should be reasonably close to 42.19)
**E. \( x \leq 45 \)** (this may offer some leeway as long as \( x < 42.19 \))
So the two correct selections are **D** and **E**.
Additionally, the company wishes to keep the production cost below $30,000. Thus, we set up the inequality:
\[
C(x) < 30
\]
Substituting the cost function into the inequality gives us:
\[
-0.16x^2 + 6.4x + 36 < 30
\]
Simplifying this:
\[
-0.16x^2 + 6.4x + 6 < 0
\]
To solve for \( x \), we can rearrange the equation:
\[
-0.16x^2 + 6.4x + 6 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.16 \), \( b = 6.4 \), and \( c = 6 \), we find the roots.
Calculating the discriminant:
\[
D = b^2 - 4ac = (6.4)^2 - 4(-0.16)(6) = 40.96 + 9.6 = 50.56
\]
Finding the roots:
\[
x = \frac{-6.4 \pm \sqrt{50.56}}{2(-0.16)}
\]
Calculating:
\[
\sqrt{50.56} \approx 7.1
\]
This leads to two roots:
\[
x = \frac{-6.4 + 7.1}{-0.32} \text{and} x = \frac{-6.4 - 7.1}{-0.32}
\]
Calculating the first root:
\[
x_1 \approx \frac{0.7}{-0.32} \approx -2.19
\]
Calculating the second root:
\[
x_2 \approx \frac{-13.5}{-0.32} \approx 42.19
\]
Evaluating the inequality \( C(x) < 30 \), we find that \( x \) must be in the range:
\[
-2.19 < x < 42.19
\]
Since \( x \) has to be greater than or equal to zero, we further restrict our domain to:
\[
0 \leq x < 42.19
\]
The reasonable constraints for the model, therefore, are:
1. \( x \geq 0 \)
2. From the roots, we have \( x < 42.19 \), or rounded, \( x \leq 42 \) or similar.
Based on the options presented, the correct constraints from the given options should be:
**D. \( x < 40.92 \)** (since it should be reasonably close to 42.19)
**E. \( x \leq 45 \)** (this may offer some leeway as long as \( x < 42.19 \))
So the two correct selections are **D** and **E**.