To determine the area of the dilated rectangle, we can compare the areas of the original rectangle and the dilated rectangle.
-
Original Rectangle: The area \( x \) of the original rectangle is calculated as follows: \[ x = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2 \]
-
Dilated Rectangle: The area of the dilated rectangle can be calculated similarly: \[ \text{Area of Dilated Rectangle} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2 \]
Now, to relate the area of the dilated rectangle to the area of the original rectangle, we can find the scale factor of dilation.
- Dilation Factor: The dimensions of the rectangle have been scaled from:
- Width: \( 3 , \text{ft} \) to \( 7.5 , \text{ft} \) (scale factor of \( \frac{7.5}{3} = 2.5 \))
- Height: \( 2 , \text{ft} \) to \( 5 , \text{ft} \) (scale factor of \( \frac{5}{2} = 2.5 \))
The area of a shape changes according to the square of the scale factor when the shape is dilated. Therefore, if the scale factor is \( 2.5 \), the area of the dilated rectangle in terms of the original area \( x \) is given by:
\[ \text{Area of Dilated Rectangle} = x \times (2.5)^2 \]
From the options provided, the correct expression to find the area of the dilated rectangle is:
\[ x \left( 2.5 \right)^2 \]
So, the answer is: x(2.5)²
1 answer