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Pattie's Produce charges \$2.29 for a package of strawberries. On average, Pattie's Produce sells 95 packages of strawberries daily. They estimate that for each 20-cent increase in the cost of a package of strawberries, 9 fewer packages will be sold each day. Let [tex]$x[tex]$[/tex] represent the number of 20-cent increases in the cost of a package of strawberries.
Which inequality represents the values of [tex]$[/tex]x$[/tex] that would allow Pattie's Produce to have a daily revenue of at least \$255 from selling the packages of strawberries?
A. [tex]-1.8 x^2 - 21.61 x + 217.55 \leq 255[/tex]
B. [tex]-1.8 x^2 + 21.61 x + 217.55 \geq 255[/tex]
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]
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