s = \frac{13 + 14 + 15}{2} = 21 \text{ m.}
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- \boxed{16\pi}
- A climate change journalist observes a triangular seawall with side lengths 13 m, 14 m, and 15 m. To assess its structural integrity, she needs to determine the length of the shortest altitude. What is the length of the shortest altitude, in meters?
- First, we compute the area of the triangle using Heron’s formula. The semi-perimeter \( s \) is:
- A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}
- = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84 \text{ square meters.}
- The altitude corresponding to a side is given by \( h = \frac{2A}{\text{side}} \). We compute the altitudes for each side:
