To find the area of the dilated rectangle, we can compare the sides of the original rectangle and the dilated rectangle.
The original rectangle measures \(3 , \text{ft} \times 2 , \text{ft}\), and its area \(x\) is calculated as:
\[ x = 3 \times 2 = 6 , \text{ft}^2 \]
The dilated rectangle measures \(7.5 , \text{ft} \times 5 , \text{ft}\). The area of the dilated rectangle is:
\[ \text{Area of dilated rectangle} = 7.5 \times 5 = 37.5 , \text{ft}^2 \]
To find the scale factor of the dilation, we can calculate the ratio of the sides:
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For the length: \[ \frac{7.5}{3} = 2.5 \]
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For the width: \[ \frac{5}{2} = 2.5 \]
Both sides are scaled by a factor of \(2.5\).
The area scales by the square of the scale factor, so:
\[ \text{Area of dilated rectangle} = x \times (2.5^2) \]
As such, the expression to find the area of the dilated rectangle is:
\[ x(2.5^2) \]
From the options provided, the correct choice is:
\[ x(2.5)^2 \]
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