To find the area of the dilated rectangle, we can first calculate the area of the original rectangle and the dilated rectangle.
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The area of the original rectangle (measuring 3 ft. by 2 ft.) is: \[ x = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2 \]
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The area of the dilated rectangle (measuring 7.5 ft. by 5 ft.) is: \[ \text{Area} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2 \]
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To find the relationship between the original area and the dilated area, we observe the scale factor of the dilation. The scale factors for the width and height can be found as follows:
- Width: \(\frac{7.5}{3} = 2.5\)
- Height: \(\frac{5}{2} = 2.5\)
Since both the width and height were multiplied by the same scale factor of \(2.5\), we can deduce the area relationship: \[ \text{Area of dilated rectangle} = \text{Area of original rectangle} \times (2.5)^2 = x \times (2.5)^2 \]
So, the expression Jorge can use to discover the area of the dilated rectangle is: \[ x(2.5)^2 \]