Question

Jorge draws a rectangle measuring 3 ft. by 2 ft. Then, he dilates the rectangle so it is 7.5 ft. by 5 ft. If x represents the area of the original rectangle, which expression can he use to discover the area of the dilated rectangle?

A. x + (2.5)^2
B. x(2.5)
C. x(1.5)^2
D. x(2.5)^2

Answer 1

To find the area of the dilated rectangle, we can first calculate the area of the original rectangle and establish a relationship with the area of the dilated rectangle.

1. Calculate the area of the original rectangle:
\[
\text{Area}_{\text{original}} = \text{length} \times \text{width} = 3 \, \text{ft} \times 2 \, \text{ft} = 6 \, \text{ft}^2
\]
So, \(x = 6 \, \text{ft}^2\).

2. Calculate the area of the dilated rectangle:
\[
\text{Area}_{\text{dilated}} = \text{length} \times \text{width} = 7.5 \, \text{ft} \times 5 \, \text{ft} = 37.5 \, \text{ft}^2
\]

3. To find the scaling factor (dilation factor), we can divide the dimensions of the dilated rectangle by those of the original rectangle:
- The length dilation factor is \( \frac{7.5}{3} = 2.5 \)
- The width dilation factor is \( \frac{5}{2} = 2.5 \)

Since the rectangle is dilated uniformly, the area of the dilated rectangle is related to the area of the original rectangle by the square of the dilation factor:
\[
\text{Area}_{\text{dilated}} = \text{Area}_{\text{original}} \times ( \text{dilation factor} )^2
\]
Substituting the values:
\[
\text{Area}_{\text{dilated}} = x \times (2.5)^2
\]

Thus, the correct expression Jorge can use to discover the area of the dilated rectangle is:
\[
\boxed{D. \, x(2.5)^2}
\]