Question

t ticket: finding an area

1. a 6 inch by 10 inch rectangle is dilated by a factor of 3.

a. what is the area of the dilated rectangle? find your answer only by finding the area of the original rectangle and multiplying appropriately.
b. support your answer to part a by sketching the new rectangle and calculating its area using the traditional $a = l \cdot w$ method.

Explanation:

Part a

Step 1: Find area of original rectangle

The formula for the area of a rectangle is \( A = l \times w \), where \( l = 6 \) inches and \( w = 10 \) inches. So, \( A_{original}=6\times10 = 60 \) square inches.

Step 2: Multiply by dilation factor squared (since area scales with the square of the linear dilation factor)

The dilation factor is 3, so we multiply the original area by \( 3^2=9 \). Thus, \( A_{dilated}=60\times9 = 540 \) square inches.

Step 1: Find dimensions of dilated rectangle

When a rectangle is dilated by a factor of 3, each side length is multiplied by 3. So the new length \( l_{new}=6\times3 = 18 \) inches and the new width \( w_{new}=10\times3 = 30 \) inches.

Step 2: Calculate area using \( A = l\times w \)

Using the formula for the area of a rectangle with \( l = 18 \) inches and \( w = 30 \) inches, we get \( A = 18\times30=540 \) square inches. (Sketch: Draw a rectangle with length 18 units and width 30 units, label the sides, and then use the area formula.)

Answer:

540 square inches

Part b