Question

Use the area model to multiply
11

2
(4d+3).
First, find the partial products. Write numbers as integers, decimals, or simplified proper or improper fractions.
4d 3
11

2


Row, rectangle. Width. Height. Find the total area.. Areacolumn, missing side length.123
Now, write the product.
11

2
(4

Answer 1

To use the area model to multiply \( \frac{11}{2} (4d + 3) \), we'll break it down into parts and calculate the area of each rectangle created by the distribution.

### Step 1: Set Up the Area Model

We will multiply \( \frac{11}{2} \) by both terms inside the parentheses: \( 4d \) and \( 3 \).

**Setting up the area model:**

- One side of the rectangle will represent \( \frac{11}{2} \).
- The other side will be split into two sections: \( 4d \) and \( 3 \).

```
+-----------+-----------+
| | |
| 4d | 3 |
| | |
+-----------+-----------+
| Width | Width |
| = 11/2 | = 11/2 |
+-----------+-----------+
```

### Step 2: Calculate the Partial Products

1. **Multiply \( \frac{11}{2} \) by \( 4d \):**
\[
\frac{11}{2} \times 4d = \frac{11 \times 4d}{2} = \frac{44d}{2} = 22d
\]

2. **Multiply \( \frac{11}{2} \) by \( 3 \):**
\[
\frac{11}{2} \times 3 = \frac{11 \times 3}{2} = \frac{33}{2}
\]

### Step 3: Add the Partial Products

Now, we add the two partial products together:

\[
22d + \frac{33}{2}
\]

### Final Answer

Thus, the product of \( \frac{11}{2} (4d + 3) \) is:

\[
22d + \frac{33}{2}
\]

This represents the total area composed of two rectangles representing each part of the original expression multiplied by \( \frac{11}{2} \).