Let's break down the problem step by step to determine the equation that represents Sarah's weekly sales income [tex]\( y \)[/tex] after [tex]\( x \)[/tex] price increases of \[tex]$0.85.
1. Initial Conditions:
- Initial trays sold: \( 450 \)
- Initial price per tray: \( \$[/tex]8.00 \)
2. Effects of Price Increases:
- For every [tex]\( \$0.85 \)[/tex] increase in price, 15 fewer trays are sold.
- If [tex]\( x \)[/tex] is the number of price increases, then:
- The price per tray after [tex]\( x \)[/tex] increases will be: [tex]\( 8 + 0.85x \)[/tex]
- The number of trays sold after [tex]\( x \)[/tex] increases will be: [tex]\( 450 - 15x \)[/tex]
3. Weekly Sales Income:
- The weekly sales income [tex]\( y \)[/tex] is given by:
[tex]\[
y = (\text{price per tray}) \times (\text{number of trays sold})
\][/tex]
- Substituting the expressions for price per tray and number of trays sold into the equation for [tex]\( y \)[/tex]:
[tex]\[
y = (8 + 0.85x) \times (450 - 15x)
\][/tex]
4. Expanding the Expression:
- Multiply the terms inside the parentheses:
[tex]\[
y = (8 + 0.85x) \times (450 - 15x)
\][/tex]
[tex]\[
y = 8 \times 450 + 8 \times (-15x) + 0.85x \times 450 + 0.85x \times (-15x)
\][/tex]
- Simplify each term:
[tex]\[
8 \times 450 = 3600
\][/tex]
[tex]\[
8 \times (-15x) = -120x
\][/tex]
[tex]\[
0.85x \times 450 = 382.5x
\][/tex]
[tex]\[
0.85x \times (-15x) = -12.75x^2
\][/tex]
5. Combining Like Terms:
- Combine the linear terms involving [tex]\( x \)[/tex]:
[tex]\[
-120x + 382.5x = 262.5x
\][/tex]
- Now, put all the terms together:
[tex]\[
y = 3600 + 262.5x - 12.75x^2
\][/tex]
- Rearrange the terms to follow the standard quadratic form ([tex]\( ax^2 + bx + c \)[/tex]):
[tex]\[
y = -12.75x^2 + 262.5x + 3600
\][/tex]
Thus, the correct equation is:
[tex]\[ y = -12.75x^2 + 262.5x + 3600 \][/tex]
The correct answer is:
B. [tex]\( y = -12.75x^2 + 262.5x + 3600 \)[/tex]
