Calculating the Expense to Paint a Triangular Traffic Sign
The problem requires calculating the total cost to paint a triangular traffic sign given its side lengths and the cost per square meter.
1. Calculate the Area of the Triangle
The side lengths of the triangle are $a = 13$ m, $b = 14$ m, and $c = 15$ m. We can use Heron's formula to find the area.
- Calculate the semi-perimeter ($s$):
The semi-perimeter is half the sum of the sides.
$s = \frac{a+b+c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \text{ m}$ - Apply Heron's Formula:
Area ($A$) $= \sqrt{s(s-a)(s-b)(s-c)}$
$A = \sqrt{21(21-13)(21-14)(21-15)}$ $A = \sqrt{21 \times 8 \times 7 \times 6}$ $A = \sqrt{7056}$ $A = 84 \text{ square meters}$
2. Calculate the Total Painting Cost
The cost to paint the sign is ₹5 per square meter. The total area is 84 square meters.
Total Expense = Area $\times$ Cost per square meter
$ \text{Total Expense} = 84 \, \text{m}^2 \times ₹5/\text{m}^2 $ $ \text{Total Expense} = ₹420 $Therefore, the total expense to paint the entire sign is ₹420.
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