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