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Write your answers at the back of this paper. (Please work in pairs)

A farmer puts up a fence. The fence is made up of segments that are attached to each other. The first segment has 6 boards, as shown below, and each of the segments that attach to the next part of the fence have 5 boards each.

First Segment

Second Segment

Third Segment

Fourth Segment

a) Complete the table below:

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline Segment & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline Boards & 6 & 5 & 5 & 5 & 5 & 5 & 5 \\
\hline \begin{tabular}{c}
Total \\
Boards
\end{tabular} & 6 & 11 & 16 & 21 & 26 & 31 & 36 \\
\hline
\end{tabular}

1. What pattern do you see? Give a formula for this rule.

2. Determine the number of boards used after 20 segments.

3. If the farmer uses 136 boards, how many segments of fence has he put up?

4. Instead of using 5 boards on each new segment, the farmer decides to use 4 boards to save money.

a. Determine the total number of boards used for the second segment, third segment, and fourth segment.

b. Determine a formula for this pattern.

c. How many boards would the farmer have saved after 20 segments?



Answer :

GinnyAnswer
Sure, let's break this problem down step by step.

### Part a) Complete the table below
```
Segment | 1 | 2 | 3 | 4 | 5 | 6 | 7
--------------------------------------------
Boards | 6 | 5 | 5 | 5 | 5 | 5 | 5
--------------------------------------------
Total Boards | 6 | 11| 16| 21| 26| 31| 36
```

### 1. What pattern do you see? Give a formula for this rule?
- Pattern: The first segment has 6 boards and each subsequent segment has 5 boards.
- Formula: The total number of boards after `n` segments can be calculated as:
[tex]\[ \text{Total boards} = 6 + 5 \times (n - 1) \][/tex]
Explanation:
- For the first segment, total boards = 6
- For the second segment, total boards = 6 + 5 = 11
- For the third segment, total boards = 6 + 5 + 5 = 16
- And so on...

### 2. Determine the number of boards used after 20 segments.
Using the formula:
[tex]\[ \text{Total boards} = 6 + 5 \times (20 - 1) = 6 + 5 \times 19 = 6 + 95 = 101 \][/tex]
So, after 20 segments, the farmer uses 101 boards.

### 3. If the farmer uses 136 boards, how many segments of fence has he put up?
We use the formula and set the total boards to 136:
[tex]\[ 136 = 6 + 5 \times (n - 1) \][/tex]
Solving for `n`:
[tex]\[ 136 = 6 + 5(n - 1) \][/tex]
[tex]\[ 136 - 6 = 5(n - 1) \][/tex]
[tex]\[ 130 = 5(n - 1) \][/tex]
[tex]\[ n - 1 = 26 \][/tex]
[tex]\[ n = 27 \][/tex]
So, the farmer has put up 27 segments.

### 4. Instead of using 5 boards on each new segment, the farmer decides to use 4 boards to save money.
#### a. Determine the total number of boards used for the second segment, third segment, and fourth segment.
- For the first segment: 6 boards
- For the second segment: 4 boards
- For the third segment: 4 boards
- For the fourth segment: 4 boards

Total boards for each up to the fourth segment:
- After the 1st segment: 6 boards
- After the 2nd segment: 6 + 4 = 10 boards
- After the 3rd segment: 10 + 4 = 14 boards
- After the 4th segment: 14 + 4 = 18 boards

#### b. Determine a formula for this pattern.
The formula for the total number of boards after `n` segments, if the first segment has 6 boards and each subsequent segment has 4 boards:
[tex]\[ \text{Total boards} = 6 + 4 \times (n - 1) \][/tex]
Explanation:
- For the first segment, total boards = 6
- For the second segment, total boards = 6 + 4 = 10
- For the third segment, total boards = 6 + 4 + 4 = 14
- And so on...

#### c. How many boards would the farmer have saved after 20 segments?
Using the new formula, calculate the total boards after 20 segments:
[tex]\[ \text{Total boards} = 6 + 4 \times (20 - 1) = 6 + 4 \times 19 = 6 + 76 = 82 \][/tex]
Boards saved compared to using 5 boards per segment:
[tex]\[ \text{Boards saved} = 101 - 82 = 19 \][/tex]
So, the farmer would have saved 19 boards after 20 segments by using 4 boards for each new segment instead of 5.

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