Sure, let's analyze this problem carefully and break it down into steps to determine the correct inequality that represents the values of [tex]\( x \)[/tex] which will allow Pattie's Produce to have a daily revenue of at least [tex]\( \$ 255 \)[/tex].
1. Initial Conditions:
- Cost of one package of strawberries: [tex]\( \$ 2.29 \)[/tex]
- Number of packages sold daily: [tex]\( 95 \)[/tex]
2. Revenue Calculation:
- Initial daily revenue: [tex]\( 2.29 \times 95 \)[/tex]
3. Effect of Price Increase:
- For each [tex]\( 20 \)[/tex]-cent increase in price (\[tex]$0.20), 9 fewer packages are sold.
- Let \( x \) represent the number of \( 20 \)-cent increases.
4. Revenue Equation:
- New price per package: \( 2.29 + 0.20x \)
- New number of packages sold daily: \( 95 - 9x \)
5. Daily Revenue Expression:
- New daily revenue: \( (2.29 + 0.20x)(95 - 9x) \)
6. Inequality Form:
- We need the revenue to be at least \( \$[/tex] 255 \):
[tex]\[
(2.29 + 0.20x)(95 - 9x) \geq 255
\][/tex]
7. Simplify the Revenue Expression:
- Expand the expression:
[tex]\[
R = 2.29(95 - 9x) + 0.20x(95 - 9x)
\][/tex]
[tex]\[
R = 217.55 - 20.61x + 19x - 1.8x^2
\][/tex]
[tex]\[
R = 217.55 - 1.61x - 1.8x^2
\][/tex]
8. Inequality Form:
- Substitute the simplified revenue expression into the inequality:
[tex]\[
217.55 - 1.61x - 1.8x^2 \geq 255
\][/tex]
Rewriting it in a standard inequality form for comparison with the given options:
[tex]\[
-1.8x^2 - 1.61x + 217.55 \geq 255
\][/tex]
Therefore, the correct inequality is:
C. [tex]\(-1.8x^2 - 1.61x + 217.55 \geq 255\)[/tex]
