Question

Sarah sells fruit trays at a salad shop. Last week, she sold 450 fruit trays for $8 per tray. In previous weeks, she found that for every $0.85 increase in the price, she sold 15 fewer fruit trays.

Which equation can be used to find Sarah's weekly sales income in dollar, y, after x price increases of $0.85.
A. y=-12.75x^2+272.5x+3,600
B. y=-12.75x^2+502.5x+3,600
C. y=-12.75x^2+482.5x+3,600
D. y=-12.75x^2+262.5+3,600

Answer 1

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.

1. **Initial Information**:
- **Initial Price per Tray**: $8
- **Initial Number of Trays Sold**: 450
- **Initial Sales Income**:
\[
\text{Income} = \text{Price per Tray} \times \text{Number of Trays Sold} = 8 \times 450 = 3600
\]

2. **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:
\[
\text{New Number of Trays Sold} = 450 - 15x
\]

3. **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. \( y = -12.75x^2 + 262.5x + 3600 \)**