21. Nola already knitted seven inches of a blanket. She adds to the blanket at a rate of three inches per hour. The total length of the blanket $$ b $$ is represented by the function $$ b=3 t+7 $$, where $$ t $$ is the number of hours Nola continues to knit. Determine appropriate input values for this situation. Then complete the function table for $$ b=3 t+ $$ 7. (Example 2)

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To solve this problem, we'll start by identifying appropriate input values for the number of hours, $$ t $$, that Nola continues to knit. Since time cannot be negative, we'll consider non-negative integers for $$ t $$ (i.e., $$ t = 0, 1, 2, 3, \dots $$).

**Appropriate Input Values for $$ t $$:**
- $$ t = 0 $$ hours
- $$ t = 1 $$ hour
- $$ t = 2 $$ hours
- $$ t = 3 $$ hours
- $$ t = 4 $$ hours
- $$ t = 5 $$ hours
- *(and so on)*

**Function Table for $$ b = 3t + 7 $$:**

| Number of Hours ($$ t $$) | Length of Blanket ($$ b $$) inches |
|---------------------------|------------------------------------|
| 0                         | $$ 3(0) + 7 = 7 $$                  |
| 1                         | $$ 3(1) + 7 = 10 $$                 |
| 2                         | $$ 3(2) + 7 = 13 $$                 |
| 3                         | $$ 3(3) + 7 = 16 $$                 |
| 4                         | $$ 3(4) + 7 = 19 $$                 |
| 5                         | $$ 3(5) + 7 = 22 $$                 |
| 6                         | $$ 3(6) + 7 = 25 $$                 |
| 7                         | $$ 3(7) + 7 = 28 $$                 |
| 8                         | $$ 3(8) + 7 = 31 $$                 |
| 9                         | $$ 3(9) + 7 = 34 $$                 |
| 10                        | $$ 3(10) + 7 = 37 $$                |

**Explanation:**
- **When $$ t = 0 $$:** Nola has not added any additional inches yet, so the blanket is already 7 inches long.
- **For each subsequent hour ($$ t $$),** add 3 inches to the previous length.

**Visualization:**

Let's visualize how the length of the blanket increases over time:

| Hours Knitted ($$ t $$) | Blanket Length ($$ b $$) |
|-------------------------|--------------------------|
| 0                       | 7 inches                 |
| 1                       | 10 inches                |
| 2                       | 13 inches                |
| 3                       | 16 inches                |
| 4                       | 19 inches                |
| 5                       | 22 inches                |
| 6                       | 25 inches                |
| 7                       | 28 inches                |
| 8                       | 31 inches                |
| 9                       | 34 inches                |
| 10                      | 37 inches                |

This table allows you to predict the total length of the blanket based on the number of hours Nola continues to knit.

**Usage Example:**
- If Nola knits for **4 hours**, the length of the blanket will be:
  $$
  b = 3(4) + 7 = 12 + 7 = 19 \text{ inches}
  $$
- If she knits for **10 hours**, the blanket will be:
  $$
  b = 3(10) + 7 = 30 + 7 = 37 \text{ inches}
  $$

Feel free to extend the table for more hours as needed!
**Appropriate Input Values for $$ t $$:** - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 hours **Function Table for $$ b = 3t + 7 $$:** | Hours ($$ t $$) | Blanket Length ($$ b $$) inches | |-----------------|---------------------------------| | 0 | 7 | | 1 | 10 | | 2 | 13 | | 3 | 16 | | 4 | 19 | | 5 | 22 | | 6 | 25 | | 7 | 28 | | 8 | 31 | | 9 | 34 | | 10 | 37 | **Explanation:** - Start with 7 inches. - Add 3 inches for each hour Nola knits. **Example:** - After 4 hours: $$ 3(4) + 7 = 19 $$ inches - After 10 hours: $$ 3(10) + 7 = 37 $$ inches

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