Question

deedee draws a rectangle with sides measuring 12 inches and 30 inches. she dilates the rectangle using a scale factor of \\( \frac{1}{5} \\). what is the area of the original rectangle? what is the area of the dilated rectangle? how many times greater is the area of the original rectangle to the area of the dilated rectangle? explain your answers and show your work. (3 points)

Explanation:

Step1: Calculate area of original rectangle

The formula for the area of a rectangle is \( A = l \times w \), where \( l \) is the length and \( w \) is the width. For the original rectangle, \( l = 30 \) inches and \( w = 12 \) inches. So, \( A_{original}=30\times12 = 360 \) square inches.

Step2: Find dimensions of dilated rectangle

The scale factor is \( \frac{1}{5} \). To find the new length and width, we multiply the original dimensions by the scale factor. New length \( l_{new}=30\times\frac{1}{5}=6 \) inches, new width \( w_{new}=12\times\frac{1}{5}=\frac{12}{5} = 2.4 \) inches.

Step3: Calculate area of dilated rectangle

Using the area formula for the dilated rectangle: \( A_{dilated}=l_{new}\times w_{new}=6\times2.4 = 14.4 \) square inches (or using fractions: \( 6\times\frac{12}{5}=\frac{72}{5}=14.4 \)).

Step4: Find the ratio of areas

To find how many times greater the original area is than the dilated area, we divide the original area by the dilated area: \( \frac{A_{original}}{A_{dilated}}=\frac{360}{14.4}=25 \). (Alternatively, since the scale factor for area is the square of the scale factor for linear dimensions, \( (\frac{1}{5})^2=\frac{1}{25} \), so the original area is \( 25 \) times the dilated area.)

Answer:

• Area of original rectangle: \( 360 \) square inches.
• Area of dilated rectangle: \( 14.4 \) (or \( \frac{72}{5} \)) square inches.
• The area of the original rectangle is \( 25 \) times greater than the area of the dilated rectangle.