A painter needs to cover a triangular region 62 meters by 68 meters by 70 meters. A can of paint covers 70 square meters. How many cans will be needed?
Let s be half the sum of the three sides (100 m, in this case). If a, b and c are the side lengths of the triangle, the area is
sqrt[s(s-a)(s-b)(s-c)]
= sqrt[100*38*32*30] = 1910 m^2
1910/70 = 27.3 cans are needed
Please can you explain this maths question to me regarding trigonometry triangular
To find out how many cans are needed, we need to calculate the total area that needs to be covered by paint.
1. Calculate the area of each side of the triangular region:
- Side 1 = 62 meters
- Side 2 = 68 meters
- Side 3 = 70 meters
2. Use Heron's formula to calculate the area of the triangle, which is the sum of the areas of its three sides:
- s = (Side 1 + Side 2 + Side 3) / 2
- Area = square root of (s * (s - Side 1) * (s - Side 2) * (s - Side 3))
3. Calculate the total area to be covered by multiplying the area of the triangle by 2 (since there are two sides with the same area):
- Total Area = Area * 2
4. Divide the total area by the area covered by each can of paint to find out the number of cans needed:
- Number of Cans = Total Area / Area Covered by Each Can
Now, let's calculate the values step-by-step:
Step 1: Calculate the area of each side of the triangle.
- Area of Side 1 = 62m * 68m / 2 = 2108 square meters
- Area of Side 2 = 68m * 70m / 2 = 2380 square meters
- Area of Side 3 = 70m * 62m / 2 = 2170 square meters
Step 2: Use Heron's formula to calculate the area of the triangle.
- s = (62m + 68m + 70m) / 2 = 100m (semi-perimeter)
- Area = sqrt(100m * (100m - 62m) * (100m - 68m) * (100m - 70m)) = sqrt(100m * 38m * 32m * 30m) = sqrt(36480000m^2) ≈ 6035.53 square meters
Step 3: Calculate the total area to be covered.
- Total Area = Area * 2 = 6035.53m^2 * 2 = 12071.06 square meters
Step 4: Calculate the number of cans needed.
- Number of Cans = Total Area / Area Covered by Each Can = 12071.06 square meters / 70 square meters = 172.44 ≈ 173 cans
Therefore, the painter will need approximately 173 cans of paint to cover the triangular region.
To find out the number of cans needed, we need to calculate the total area of the triangular region and then divide it by the coverage area of a single can.
First, let's calculate the area of each side of the triangle using Heron's formula:
s = (a + b + c)/2
Area = √(s * (s-a) * (s-b) * (s-c))
where a, b, and c are the lengths of the sides of the triangle, and s is the semiperimeter.
Given:
a = 62 meters
b = 68 meters
c = 70 meters
Calculate the semiperimeter:
s = (62 + 68 + 70)/2 = 100 meters
Now, calculate the area of each side:
Area₁ = √(100 * (100-62) * (100-68) * (100-70)) ≈ 1928.2 square meters
Area₂ = √(100 * (100-62) * (100-70) * (100-68)) ≈ 1928.2 square meters
Area₃ = √(100 * (100-68) * (100-70) * (100-62)) ≈ 2029.5 square meters
Finally, add up the areas of all three sides:
Total area = Area₁ + Area₂ + Area₃ ≈ 1928.2 + 1928.2 + 2029.5 ≈ 5885.9 square meters
Now, divide the total area of the triangular region by the coverage area of a single can (70 square meters):
Number of cans needed = Total area / Coverage area = 5885.9 / 70 ≈ 84.1
So, a painter will need approximately 84 cans of paint to cover the given triangular region.