Question

DeeDee draws a rectangle with sides measuring x inches and y inches. She dilates the rectangle using a scale factor of 14 . What is the area of the original rectangle in terms of x and y? What is the area of the dilated rectangle? How many times greater is the area of the original rectangle compared to the area of the dilated rectangle? Explain your answers and show all your work.(4 points)

Answer 1

To solve the problem step by step, we need to find the area of the original rectangle, the area of the dilated rectangle, and the comparison between the two areas.

### Step 1: Area of the Original Rectangle

The area \( A \) of a rectangle is calculated using the formula:
\[
A = \text{length} \times \text{width}
\]
For DeeDee's rectangle with sides measuring \( x \) inches and \( y \) inches, the area of the original rectangle \( A_{\text{original}} \) is:
\[
A_{\text{original}} = x \cdot y
\]

### Step 2: Area of the Dilated Rectangle

When dilating the rectangle with a scale factor of 14, each side of the rectangle is multiplied by 14. Hence, the new dimensions after dilation will be:
- Length: \( 14x \)
- Width: \( 14y \)

Now, we can calculate the area of the dilated rectangle \( A_{\text{dilated}} \):
\[
A_{\text{dilated}} = \text{length}_{\text{dilated}} \times \text{width}_{\text{dilated}} = (14x) \cdot (14y)
\]
\[
A_{\text{dilated}} = 196xy
\]

### Step 3: Comparison of Areas

Next, we want to compare the area of the original rectangle to the area of the dilated rectangle. We can set up a ratio:
\[
\frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{196xy}{xy} = 196
\]

This means that the area of the dilated rectangle is **196 times greater** than the area of the original rectangle.

### Summary of the Answers
- Area of the original rectangle: \( x \cdot y \)
- Area of the dilated rectangle: \( 196xy \)
- The area of the dilated rectangle is 196 times greater than the area of the original rectangle.

### Conclusion

1. The area of the original rectangle is \( xy \) square inches.
2. The area of the dilated rectangle is \( 196xy \) square inches.
3. The area of the dilated rectangle is \( 196 \) times greater than the area of the original rectangle.