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Sagot :
Alright, let's solve this step by step!
### Step 1: Given Information
1. The cost of sowing the field is Rs. 783.
2. The cost per hectare is Rs. 58.
3. The base of the triangular field is three times its altitude.
### Step 2: Calculate the Area of the Field in Hectares
We start by determining the area of the field in hectares.
[tex]\[ \text{Area in hectares} = \frac{\text{Total cost}}{\text{Cost per hectare}} \][/tex]
[tex]\[ \text{Area in hectares} = \frac{783 \text{ Rs}}{58 \text{ Rs per hectare}} \][/tex]
[tex]\[ \text{Area in hectares} = 13.5 \][/tex]
### Step 3: Convert Hectares to Square Meters
Since 1 hectare is equal to 10,000 square meters, we need to convert the area from hectares to square meters.
[tex]\[ \text{Area in square meters} = \text{Area in hectares} \times 10,000 \][/tex]
[tex]\[ \text{Area in square meters} = 13.5 \times 10,000 \][/tex]
[tex]\[ \text{Area in square meters} = 135,000 \][/tex]
### Step 4: Use the Formula for the Area of a Triangle
The area of a triangle is given by:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{altitude} \][/tex]
It is given that the base ([tex]\(b\)[/tex]) is three times the altitude ([tex]\(h\)[/tex]).
[tex]\[ b = 3h \][/tex]
Substitute [tex]\( b = 3h \)[/tex] into the area formula:
[tex]\[ 135,000 = \frac{1}{2} \times 3h \times h \][/tex]
[tex]\[ 135,000 = \frac{3}{2} h^2 \][/tex]
### Step 5: Solve for the Altitude ([tex]\( h \)[/tex])
To find [tex]\( h \)[/tex], multiply both sides by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ h^2 = \frac{2}{3} \times 135,000 \][/tex]
[tex]\[ h^2 = 90,000 \][/tex]
Take the square root of both sides:
[tex]\[ h = \sqrt{90,000} \][/tex]
[tex]\[ h = 300 \text{ meters} \][/tex]
### Step 6: Find the Base ([tex]\( b \)[/tex])
Since the base is three times the altitude:
[tex]\[ b = 3h \][/tex]
[tex]\[ b = 3 \times 300 \][/tex]
[tex]\[ b = 900 \text{ meters} \][/tex]
### Final Answer
The altitude ([tex]\(h\)[/tex]) of the triangular field is [tex]\(300\)[/tex] meters, and the base ([tex]\(b\)[/tex]) is [tex]\(900\)[/tex] meters.
### Step 1: Given Information
1. The cost of sowing the field is Rs. 783.
2. The cost per hectare is Rs. 58.
3. The base of the triangular field is three times its altitude.
### Step 2: Calculate the Area of the Field in Hectares
We start by determining the area of the field in hectares.
[tex]\[ \text{Area in hectares} = \frac{\text{Total cost}}{\text{Cost per hectare}} \][/tex]
[tex]\[ \text{Area in hectares} = \frac{783 \text{ Rs}}{58 \text{ Rs per hectare}} \][/tex]
[tex]\[ \text{Area in hectares} = 13.5 \][/tex]
### Step 3: Convert Hectares to Square Meters
Since 1 hectare is equal to 10,000 square meters, we need to convert the area from hectares to square meters.
[tex]\[ \text{Area in square meters} = \text{Area in hectares} \times 10,000 \][/tex]
[tex]\[ \text{Area in square meters} = 13.5 \times 10,000 \][/tex]
[tex]\[ \text{Area in square meters} = 135,000 \][/tex]
### Step 4: Use the Formula for the Area of a Triangle
The area of a triangle is given by:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{altitude} \][/tex]
It is given that the base ([tex]\(b\)[/tex]) is three times the altitude ([tex]\(h\)[/tex]).
[tex]\[ b = 3h \][/tex]
Substitute [tex]\( b = 3h \)[/tex] into the area formula:
[tex]\[ 135,000 = \frac{1}{2} \times 3h \times h \][/tex]
[tex]\[ 135,000 = \frac{3}{2} h^2 \][/tex]
### Step 5: Solve for the Altitude ([tex]\( h \)[/tex])
To find [tex]\( h \)[/tex], multiply both sides by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ h^2 = \frac{2}{3} \times 135,000 \][/tex]
[tex]\[ h^2 = 90,000 \][/tex]
Take the square root of both sides:
[tex]\[ h = \sqrt{90,000} \][/tex]
[tex]\[ h = 300 \text{ meters} \][/tex]
### Step 6: Find the Base ([tex]\( b \)[/tex])
Since the base is three times the altitude:
[tex]\[ b = 3h \][/tex]
[tex]\[ b = 3 \times 300 \][/tex]
[tex]\[ b = 900 \text{ meters} \][/tex]
### Final Answer
The altitude ([tex]\(h\)[/tex]) of the triangular field is [tex]\(300\)[/tex] meters, and the base ([tex]\(b\)[/tex]) is [tex]\(900\)[/tex] meters.
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