Question: 2 6 Using FOIL to Represent Area
LEARNING OBJECTIVE: Calculate area by using binomial multiplication.
Ralph, Jennifer, and Ted want to build a garden with flowers, herbs, and vegetables. The herbs are as wide as the flowers, and 4 feet long. The vegetable garden is enlarged to yield the layout below.
Select the expression that represents the total garden area.
center: Three yellow circles are arranged horizontally in the center of the page, with a blue left chevron to the left of the circles and a blue right chevron to the right of the circles.
LEARNING OBJECTIVE: Calculate area by using binomial multiplication.
Ralph, Jennifer, and Ted want to build a garden with flowers, herbs, and vegetables. The herbs are as wide as the flowers, and 4 feet long. The vegetable garden is enlarged to yield the layout below.
Select the expression that represents the total garden area.
4
3x
8
x
Herbs rectangle: top-left, width 8 and height 4.
Flowers rectangle: bottom-left, width 8 and height 3x.
Vegetables rectangle: right side, width x and height 4 + 3x.
Overall garden shape is a large rectangle with total width 8 + x and total height 4 + 3x.
1. **State the problem:** We need to find the total area of the garden composed of three parts: herbs, flowers, and vegetables.
2. **Identify dimensions:**
- Herbs rectangle: width $8$, height $4$.
- Flowers rectangle: width $8$, height $3x$.
- Vegetables rectangle: width $x$, height $4 + 3x$.
3. **Overall garden dimensions:** The garden forms a large rectangle with width $8 + x$ and height $4 + 3x$.
4. **Formula for area of a rectangle:**
$$\text{Area} = \text{width} \times \text{height}$$
5. **Calculate total garden area:**
$$\text{Area} = (8 + x)(4 + 3x)$$
6. **Use FOIL to expand:**
$$
(8 + x)(4 + 3x) = 8 \times 4 + 8 \times 3x + x \times 4 + x \times 3x
$$
$$
= 32 + 24x + 4x + 3x^2
$$
7. **Combine like terms:**
$$
32 + (24x + 4x) + 3x^2 = 32 + 28x + 3x^2
$$
8. **Final expression:**
$$\boxed{3x^2 + 28x + 32}$$
This expression represents the total area of the garden in square feet.