Question

The following figure is a rectangle made up of two smaller rectangles.
Top of the dark rectangle : x
Top of the light rectangle: 5
Side of the dark rectangle: 6
Side of the light rectangle: 6
(a) Find the area of the following (in square units).
Thedarkrectangle(onthe left):_
The light rectangle(onthe 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: _

Answer 1

To find the areas of the dark and light rectangles and the entire figure, we can follow these steps:

### Given Information
1. **Top of the dark rectangle (height)**: \( x \) (we'll assume it's a variable for now)
2. **Top of the light rectangle (height)**: 5
3. **Side of the dark rectangle (width)**: 6
4. **Side of the light rectangle (width)**: 6

### Areas of the Rectangles

#### (a) Find the areas of the rectangles

1. **Area of the dark rectangle**:
\[
\text{Area}_{\text{dark}} = \text{height} \times \text{width} = x \times 6 = 6x \quad \text{(square units)}
\]

2. **Area of the light rectangle**:
\[
\text{Area}_{\text{light}} = \text{height} \times \text{width} = 5 \times 6 = 30 \quad \text{(square units)}
\]

### Summary of Areas
- The area of the dark rectangle: \( 6x \) square units
- The area of the light rectangle: \( 30 \) square units

#### (b) The area of the entire figure

1. **As a sum of two areas**:
\[
\text{Area}_{\text{total}} = \text{Area}_{\text{dark}} + \text{Area}_{\text{light}} = 6x + 30 \quad \text{(square units)}
\]

2. **As a product of the length and width**:
- The total height of the entire figure is \( x + 5 \).
- The total width remains \( 6 + 6 = 12 \).
\[
\text{Area}_{\text{total}} = \text{height}_{\text{total}} \times \text{width}_{\text{total}} = (x + 5) \times 12 \quad \text{(square units)}
\]

### Final Results
- Area of the dark rectangle: \( 6x \)
- Area of the light rectangle: \( 30 \)
- Area of the entire figure (sum): \( 6x + 30 \)
- Area of the entire figure (product): \( 12(x + 5) \)

These two expressions for the total area are equivalent since we can expand the product:
\[
12(x + 5) = 12x + 60
\]
So the two representations for the total area hold true.