Question

The following figure is a rectangle made up of two smaller rectangles.
7
7
2
x
(a) Find the area of the following (in square units).
Thedarkrectangle(ontheleft):
Thelightrectangle(ontheright):
(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:

Answer 1

To solve this problem, we first need to define the dimensions of the two rectangles based on the information provided.

1. **Dimensions of the dark rectangle (on the left)**:
- Height = 7 units (as stated)
- Width = 2 units (as stated)

Thus, the area \( A_d \) of the dark rectangle is:
\[
A_d = \text{Height} \times \text{Width} = 7 \times 2 = 14 \text{ square units}
\]

2. **Dimensions of the light rectangle (on the right)**:
- Height = 7 units (same height as the dark rectangle)
- Width = \( x \) units (variable width given in the problem)

Thus, the area \( A_l \) of the light rectangle is:
\[
A_l = \text{Height} \times \text{Width} = 7 \times x = 7x \text{ square units}
\]

### (a) Find the area of the following (in square units).
- **Area of the dark rectangle (on the left)**:
\[
A_d = 14 \text{ square units}
\]
- **Area of the light rectangle (on the right)**:
\[
A_l = 7x \text{ square units}
\]

### (b) Give the area of the entire figure (in square units) in two different ways.

1. **As a sum of two areas**:
The total area \( A_t \) of the entire figure is the sum of the areas of the two rectangles:
\[
A_t = A_d + A_l = 14 + 7x \text{ square units}
\]

2. **As a product of the length and width**:
The entire figure is a rectangle with:
- Height = 7 units (common height for both rectangles)
- Width = \( 2 + x \) (the width of the dark rectangle plus the width of the light rectangle)

Thus, the area can also be calculated as:
\[
A_t = \text{Height} \times \text{Width} = 7 \times (2 + x) = 14 + 7x \text{ square units}
\]

In summary, the total area of the entire figure from both methods is given by:
- As a sum: \( A_t = 14 + 7x \)
- As a product: \( A_t = 14 + 7x \)

Both approaches yield the same total area.