Let's break down the problem step-by-step.
1. Initial Parameters:
- Initial cost per package: [tex]\( \$2.29 \)[/tex]
- Initial number of packages sold: [tex]\( 95 \)[/tex]
- Revenue target: [tex]\( \$255 \)[/tex]
- Increase per package for each [tex]\( 20 \)[/tex]-cent increase: [tex]\( \$0.20 \)[/tex]
- Decrease in packages sold for each [tex]\( 20 \)[/tex]-cent increase: [tex]\( 9 \)[/tex] packages
2. Revenue Equation:
- Let [tex]\( x \)[/tex] be the number of [tex]\( 20 \)[/tex]-cent increases.
- New cost per package after [tex]\( x \)[/tex] [tex]\( 20 \)[/tex]-cent increases: [tex]\( 2.29 + 0.20x \)[/tex]
- New number of packages sold after [tex]\( x \)[/tex] [tex]\( 20 \)[/tex]-cent increases: [tex]\( 95 - 9x \)[/tex]
- Revenue after [tex]\( x \)[/tex] increases: [tex]\( (2.29 + 0.20x) \times (95 - 9x) \)[/tex]
3. Revenue Inequality:
- We need the revenue to be at least [tex]\( 255 \)[/tex], so the inequality is:
[tex]\[
(2.29 + 0.20x) \times (95 - 9x) \geq 255
\][/tex]
4. Formulating and Simplifying the Inequality:
The left-hand side of the inequality represents the revenue:
[tex]\[
(2.29 + 0.20x) \times (95 - 9x)
\][/tex]
Expanding and simplifying it would give us a quadratic expression:
[tex]\[
2.29 \times 95 + 2.29 \times (-9x) + 0.20x \times 95 + 0.20x \times (-9x)
\][/tex]
[tex]\[
= 217.55 - 20.61x + 19x - 1.8x^2
\][/tex]
[tex]\[
= 217.55 - 1.61x - 1.8x^2
\][/tex]
So, the inequality becomes:
[tex]\[
217.55 - 1.61x - 1.8x^2 \geq 255
\][/tex]
Rewriting it in standard form:
[tex]\[
- 1.8x^2 - 1.61x + 217.55 \geq 255
\][/tex]
Subtracting 255 from both sides:
[tex]\[
- 1.8x^2 - 1.61x + 217.55 - 255 \geq 0
\][/tex]
[tex]\[
- 1.8x^2 - 1.61x - 37.45 \geq 0
\][/tex]
Finally, the inequality can be expressed as:
[tex]\[
-1.8x^2 - 1.61x \leq -37.45
\][/tex]
Comparing to the given option, we find that the inequality most closely matches option D:
[tex]\[
-1.8x^2 + 1.61x + 217.55 \leq 255
\][/tex]
Given this, the correct answer is:
D. [tex]\( -1.8 x^2 + 1.61 x + 217.55 \leq 255\)[/tex]
