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📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question:** A soil scientist studying erosion patterns near a river observes a triangular plot of land with side lengths of 13 m, 14 m, and 15 m. To analyze drainage flow, she needs the length of the shortest altitude. What is the length of the shortest altitude?

📌 Solution:** First, compute the area of the triangle using Heron’s formula. The semi-perimeter $s$ is:

📌 = \sqrt{7056} = 84

📌 The area is 84 m². The shortest altitude corresponds to the longest side, which is 15 m. Using $A = \frac{1}{2} \times \text{base} \times \text{height}$:

📌 = \frac{1}{2} \times 15 \times h \Rightarrow h = \frac{84 \times 2}{15} = \frac{168}{15} = 11.2

📌 Thus, the shortest altitude is $\boxed{11.2}$ meters.

📌 Question:** An archaeologist uses ground-penetrating radar to map a circular foundation of an ancient hut. A rectangular structure, 6 m by 8 m, is perfectly inscribed in the circular base. What is the circumference of the circle in meters? Express your answer in terms of $\pi$.

📌 Solution:** Since the rectangle is inscribed in the circle, its diagonal is the diameter of the circle. Using the Pythagorean theorem:

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