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?

Answer 1

To find the area of a rectangle, you multiply its length by its width.

First, let's determine the area of the original rectangle that measures 3 ft by 2 ft:

\[
x = \text{length} \times \text{width} = 3 \, \text{ft} \times 2 \, \text{ft} = 6 \, \text{ft}^2.
\]

Next, we calculate the area of the dilated rectangle that measures 7.5 ft by 5 ft:

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

To relate the areas, Jorge can express the area of the dilated rectangle in terms of the area of the original rectangle. If \( x \) represents the area of the original rectangle (\( x = 6 \, \text{ft}^2 \)), we can express the area of the dilated rectangle as a multiple of the area of the original rectangle.

The scale factor of the dilation can be determined by comparing the dimensions of the rectangles:

- The length scale factor: \( \frac{7.5 \, \text{ft}}{3 \, \text{ft}} = 2.5 \)
- The width scale factor: \( \frac{5 \, \text{ft}}{2 \, \text{ft}} = 2.5 \)

Since both dimensions have the same scale factor, we confirm that the rectangle was uniformly dilated by a factor of 2.5.

The area scale factor is the square of the linear scale factor:

\[
\text{Area scale factor} = (2.5)^2 = 6.25.
\]

Therefore, the area of the dilated rectangle can be expressed in terms of the original rectangle's area \( x \):

\[
\text{Area of dilated rectangle} = 6.25 \times x.
\]

Thus, the expression Jorge can use to discover the area of the dilated rectangle is:

\[
6.25x.
\]