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

1 answer

First, we can find the area of the original rectangle. The area \( x \) of the original rectangle, which measures 3 ft by 2 ft, is calculated as:

\[ x = 3 \times 2 = 6 \text{ square feet} \]

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

\[ \text{Area of the dilated rectangle} = 7.5 \times 5 = 37.5 \text{ square feet} \]

To relate the area of the dilated rectangle to the area of the original rectangle using a dilation factor, we first determine the scale factor. The width goes from 3 ft to 7.5 ft, and the height goes from 2 ft to 5 ft. We can find the dilation scale factor for either dimension:

\[ \text{Scale factor for width} = \frac{7.5}{3} = 2.5 \] \[ \text{Scale factor for height} = \frac{5}{2} = 2.5 \]

Since the scale factors are equal, the dilation factor is 2.5.

When a geometric shape is dilated by a scale factor \( k \), the area changes by a factor of \( k^2 \). Thus, the area of the dilated rectangle can be related to the area of the original rectangle by:

\[ \text{Area of the dilated rectangle} = \text{Area of the original rectangle} \times k^2 \]

Substituting in the scale factor \( k = 2.5 \):

\[ \text{Area of the dilated rectangle} = x \times (2.5)^2 \]

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

\(\boxed{A}\) \( x(2.5)^2 \)