Of course! Let's break down this problem step-by-step to calculate the costs associated with fencing a rectangular farm plot of 100 meters by 80 meters, leaving a 4-meter gate, and using four strands of barbed wire.
1. Calculate the Perimeter of the Farm:
The farm measures 100 meters by 80 meters.
[tex]\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width}) = 2 \times (100 \, \text{m} + 80 \, \text{m}) = 2 \times 180 \, \text{m} = 360 \, \text{m}
\][/tex]
2. Adjust the Perimeter to Exclude the Gate:
The gate is 4 meters wide.
[tex]\[
\text{Perimeter to Fence} = \text{Perimeter} - \text{Gate Width} = 360 \, \text{m} - 4 \, \text{m} = 356 \, \text{m}
\][/tex]
3. Calculate the Number of Fence Posts Required:
Fence posts are placed at 4-meter intervals. Since the posts begin at one point and each interval includes a post, we add one more post.
[tex]\[
\text{Number of Posts} = \frac{\text{Perimeter to Fence}}{\text{Post Interval}} + 1 = \frac{356 \, \text{m}}{4 \, \text{m}} + 1 = 89 + 1 = 90
\][/tex]
4. Calculate the Total Cost for the Posts:
Each post costs 125 sh.
[tex]\[
\text{Total Cost for Posts} = (\text{Number of Posts}) \times (\text{Cost per Post}) = 90 \times 125 = 11250 \, \text{sh}
\][/tex]
5. Calculate the Total Wire Length Required:
The wire needs to make four complete strands around the fenced perimeter.
[tex]\[
\text{Total Wire Length} = \text{Perimeter to Fence} \times \text{Number of Strands} = 356 \, \text{m} \times 4 = 1424 \, \text{m}
\][/tex]
6. Calculate the Number of Wire Rolls Needed:
Each roll is 60 meters long.
[tex]\[
\text{Number of Rolls} = \frac{\text{Total Wire Length}}{\text{Length per Roll}} = \frac{1424 \, \text{m}}{60 \, \text{m}} \approx 23.733
\][/tex]
The number of rolls must be rounded up because you can't purchase a fraction of a roll. Therefore, we need 24 rolls.
7. Calculate the Total Cost for Wire Rolls:
Each roll costs 1500 sh.
[tex]\[
\text{Total Cost for Wire Rolls} = (\text{Number of Rolls}) \times (\text{Cost per Roll}) = 24 \times 1500 = 36000 \, \text{sh}
\][/tex]
8. Calculate the Total Cost for the Entire Fencing Project:
Add the cost of the wood posts, the price of the gate itself (1250 sh), and the wire rolls.
[tex]\[
\text{Total Cost} = \text{Total Cost for Posts} + \text{Total Cost for Wire Rolls} + \text{Gate Price} = 11250 \, \text{sh} + 36000 \, \text{sh} + 1250 \, \text{sh} = 48500 \, \text{sh}
\][/tex]
Thus, the step-by-step solution to the farmer's problem of calculating the total cost of fencing the plot, including the posts, wire rolls, and gate, is as follows:
- Perimeter to fence (excluding the gate): 356 meters
- Number of posts required: 90 posts
- Total cost for posts: 11250 sh
- Total wire length needed: 1424 meters
- Number of wire rolls needed: 24 rolls
- Total cost for wire rolls: 36000 sh
- Total cost for the entire fencing project: 48500 sh
