Question

the following figure is a rectangle made up of two smaller rectangles.

figure: left rectangle (dark) with width 2, height 9; right rectangle (light) with width x, height 9

(a) find the area of the following (in square units).
the dark rectangle (on the left):
the light rectangle (on the right):
(b) give the area of the entire figure (in square units) in two different ways.
as a sum of two areas:
as a product of the length and width:

Explanation:

Part (a)

Dark Rectangle (Left)

Step1: Recall area of rectangle formula

The area of a rectangle is given by \( A = \text{length} \times \text{width} \). For the dark rectangle, the length is 9 and the width is 2.

Step2: Calculate the area

Using the formula \( A = 9 \times 2 \), we get \( A = 18 \).

Light Rectangle (Right)

Step1: Recall area of rectangle formula

The area of a rectangle is \( A = \text{length} \times \text{width} \). For the light rectangle, the length is 9 and the width is \( x \).

Step2: Calculate the area

Using the formula \( A = 9 \times x \), we get \( A = 9x \).

Part (b)

As a sum of two areas

Step1: Recall the areas of the two rectangles

We found the area of the dark rectangle is 18 and the light rectangle is \( 9x \).

Step2: Sum the two areas

The total area as a sum is \( 18 + 9x \).

As a product of length and width

Answer:

Step1: Find the total length of the entire rectangle

The total length of the entire rectangle is the sum of the widths of the two smaller rectangles, which is \( 2 + x \). The height (length) of the entire rectangle is 9.

Step2: Calculate the area using the formula

Using the area formula for a rectangle \( A = \text{length} \times \text{width} \), where length is 9 and width is \( 2 + x \), we get \( A = 9(2 + x) \).

Final Answers

(a) Dark rectangle: \(\boldsymbol{18}\); Light rectangle: \(\boldsymbol{9x}\)
(b) Sum of two areas: \(\boldsymbol{18 + 9x}\); Product of length and width: \(\boldsymbol{9(2 + x)}\)