Question

the figure shows two rectangles. which expression represents the area of the shaded region?
○ 16x²
○ 34x²
○ 54x²
○ 88x²

Explanation:

Answer:

To find the area of the shaded region, we calculate the area of the larger rectangle and subtract the area of the smaller (unshaded) rectangle.

Step 1: Dimensions of the larger rectangle

• The length of the larger rectangle: \( 6x + x + x = 8x \) (adding the two \( x \)-widths to the length of the smaller rectangle).
• The width of the larger rectangle: \( 9x + x + x = 11x \) (adding the two \( x \)-widths to the width of the smaller rectangle).

Step 2: Area of the larger rectangle

\[
\text{Area}_{\text{larger}} = \text{length} \times \text{width} = 8x \times 11x = 88x^2
\]

Step 3: Area of the smaller (unshaded) rectangle

\[
\text{Area}_{\text{smaller}} = 6x \times 9x = 54x^2
\]

Step 4: Area of the shaded region

Subtract the area of the smaller rectangle from the larger one:
\[
\text{Area}_{\text{shaded}} = 88x^2 - 54x^2 = 34x^2
\]

The expression representing the area of the shaded region is \( \boldsymbol{34x^2} \), corresponding to the option:

\( \boldsymbol{34x^2} \) (the second option in the list).