Sure, let's solve this step by step.
Step 1: Identify the sides of the triangle and the rate for leveling the ground.
The given sides of the triangle are:
- [tex]\( a = 26 \)[/tex] meters
- [tex]\( b = 28 \)[/tex] meters
- [tex]\( c = 30 \)[/tex] meters
The rate for levelling the ground is ₹ 3 per square meter.
Step 2: Calculate the semi-perimeter of the triangle.
The semi-perimeter [tex]\( s \)[/tex] is calculated as:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Substituting the given values:
[tex]\[ s = \frac{26 + 28 + 30}{2} \][/tex]
[tex]\[ s = \frac{84}{2} \][/tex]
[tex]\[ s = 42 \][/tex]
So, the semi-perimeter [tex]\( s \)[/tex] is 42 meters.
Step 3: Calculate the area of the triangle using Heron's formula.
The formula for the area [tex]\( A \)[/tex] of a triangle using Heron's formula is:
[tex]\[ A = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
Let's substitute the values we have:
[tex]\[ A = \sqrt{42(42-26)(42-28)(42-30)} \][/tex]
[tex]\[ A = \sqrt{42 \times 16 \times 14 \times 12} \][/tex]
[tex]\[ A = \sqrt{112896} \][/tex]
[tex]\[ A = 336 \][/tex]
So, the area [tex]\( A \)[/tex] of the triangle is 336 square meters.
Step 4: Calculate the cost of leveling the ground.
The cost is given by multiplying the area by the rate per square meter:
[tex]\[ \text{Cost} = \text{Area} \times \text{Rate per square meter} \][/tex]
[tex]\[ \text{Cost} = 336 \times 3 \][/tex]
[tex]\[ \text{Cost} = 1008 \][/tex]
Therefore, the cost of levelling the ground is ₹ 1008.
