To determine the correct equation for Sarah's weekly sales income after \( x \) price increases of $0.85, we need to analyze the relationships involved.
-
Initial Information:
- Initial Price per Tray: $8
- Initial Number of Trays Sold: 450
- Initial Sales Income: \[ \text{Income} = \text{Price per Tray} \times \text{Number of Trays Sold} = 8 \times 450 = 3600 \]
-
Effects of Price Increases:
-
New Price after \( x \) increases: Each increase is $0.85, so after \( x \) price increases: \[ \text{New Price} = 8 + 0.85x \]
-
Decrease in Number of Trays Sold: For every $0.85 increase, Sarah sells 15 fewer trays. After \( x \) increases: \[ \text{New Number of Trays Sold} = 450 - 15x \]
-
-
Sales Income after price increases: \[ y = (\text{New Price}) \times (\text{New Number of Trays Sold}) = (8 + 0.85x)(450 - 15x) \]
Expanding this expression: \[ y = (8)(450 - 15x) + (0.85x)(450 - 15x) \] \[ y = 3600 - 120x + 382.5x - 12.75x^2 \] \[ y = 3600 + (382.5 - 120)x - 12.75x^2 \] \[ y = 3600 + 262.5x - 12.75x^2 \]
After simplifying this, the equation for \( y \) in terms of \( x \) is: \[ y = -12.75x^2 + 262.5x + 3600 \]
Now, comparing with the answer choices:
- A. \( y = -12.75x^2 + 272.5x + 3600 \)
- B. \( y = -12.75x^2 + 502.5x + 3600 \)
- C. \( y = -12.75x^2 + 482.5x + 3600 \)
- D. \( y = -12.75x^2 + 262.5x + 3600 \)
The correct equation is: D. \( y = -12.75x^2 + 262.5x + 3600 \)