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Sagot :
Certainly! Let's break down the solution step-by-step for each of the parts (a), (b), (c), and (d) given the information:
### (a) Lengths of the sides of the triangular field
The sides of the triangular field are in the ratio 2:3:4, and the perimeter of the triangle is 900 meters. Let's denote the sides of the triangle as [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
First, we need to determine the total ratio sum:
[tex]\[ 2 + 3 + 4 = 9 \][/tex]
Now we can calculate each side:
[tex]\[ a = \frac{2}{9} \times 900 = 200 \text{ meters} \][/tex]
[tex]\[ b = \frac{3}{9} \times 900 = 300 \text{ meters} \][/tex]
[tex]\[ c = \frac{4}{9} \times 900 = 400 \text{ meters} \][/tex]
### (b) Semi-perimeter of the triangle
The semi-perimeter (s) of the triangle is given by:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Substituting the values we found:
[tex]\[ s = \frac{200 + 300 + 400}{2} = 450 \text{ meters} \][/tex]
### (c) Area of the triangle using Heron's formula
Heron's formula for the area (A) of a triangle is:
[tex]\[ A = \sqrt{s(s - a)(s - b)(s - c)} \][/tex]
Substituting the values we have:
[tex]\[ A = \sqrt{450 \times (450 - 200) \times (450 - 300) \times (450 - 400)} \][/tex]
Calculating inside the square root:
[tex]\[ A = \sqrt{450 \times 250 \times 150 \times 50} \][/tex]
[tex]\[ A \approx 29047.38 \text{ square meters} \][/tex]
### (d) Total cost of fencing the land five times
The cost of fencing is Rs 200 per meter. The total cost for fencing the perimeter once is:
[tex]\[ \text{Cost per fence} = 900 \text{ meters} \times 200 \text{ Rs/meter} = 180000 \text{ Rs} \][/tex]
Since we need to fence the land five times, the total cost will be:
[tex]\[ \text{Total cost} = 5 \times 180000 = 900000 \text{ Rs} \][/tex]
### Summary
1. Lengths of the sides are:
- [tex]\(a = 200\)[/tex] meters
- [tex]\(b = 300\)[/tex] meters
- [tex]\(c = 400\)[/tex] meters
2. Semi-perimeter of the triangle is 450 meters.
3. Area of the triangle is approximately 29047.38 square meters.
4. Total cost for fencing the land five times is 900000 Rs.
### (a) Lengths of the sides of the triangular field
The sides of the triangular field are in the ratio 2:3:4, and the perimeter of the triangle is 900 meters. Let's denote the sides of the triangle as [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
First, we need to determine the total ratio sum:
[tex]\[ 2 + 3 + 4 = 9 \][/tex]
Now we can calculate each side:
[tex]\[ a = \frac{2}{9} \times 900 = 200 \text{ meters} \][/tex]
[tex]\[ b = \frac{3}{9} \times 900 = 300 \text{ meters} \][/tex]
[tex]\[ c = \frac{4}{9} \times 900 = 400 \text{ meters} \][/tex]
### (b) Semi-perimeter of the triangle
The semi-perimeter (s) of the triangle is given by:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Substituting the values we found:
[tex]\[ s = \frac{200 + 300 + 400}{2} = 450 \text{ meters} \][/tex]
### (c) Area of the triangle using Heron's formula
Heron's formula for the area (A) of a triangle is:
[tex]\[ A = \sqrt{s(s - a)(s - b)(s - c)} \][/tex]
Substituting the values we have:
[tex]\[ A = \sqrt{450 \times (450 - 200) \times (450 - 300) \times (450 - 400)} \][/tex]
Calculating inside the square root:
[tex]\[ A = \sqrt{450 \times 250 \times 150 \times 50} \][/tex]
[tex]\[ A \approx 29047.38 \text{ square meters} \][/tex]
### (d) Total cost of fencing the land five times
The cost of fencing is Rs 200 per meter. The total cost for fencing the perimeter once is:
[tex]\[ \text{Cost per fence} = 900 \text{ meters} \times 200 \text{ Rs/meter} = 180000 \text{ Rs} \][/tex]
Since we need to fence the land five times, the total cost will be:
[tex]\[ \text{Total cost} = 5 \times 180000 = 900000 \text{ Rs} \][/tex]
### Summary
1. Lengths of the sides are:
- [tex]\(a = 200\)[/tex] meters
- [tex]\(b = 300\)[/tex] meters
- [tex]\(c = 400\)[/tex] meters
2. Semi-perimeter of the triangle is 450 meters.
3. Area of the triangle is approximately 29047.38 square meters.
4. Total cost for fencing the land five times is 900000 Rs.
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