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Home » We combine the expression into a single sine function: - AssociationVoting

We combine the expression into a single sine function: - AssociationVoting

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 5Question: A zoologist measures a triangular region in the Amazon with side lengths 13 km, 14 km, and 15 km. What is the length of the shortest altitude in this region?

📌 Solution: First, calculate the area of the triangle using Heron's formula. The semi-perimeter $ s = \frac{13 + 14 + 15}{2} = 21 $. The area is $ \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84 \, \text{km}^2 $. The altitudes correspond to each side: $ h_a = \frac{2 \times 84}{13} \approx 12.92 $, $ h_b = \frac{2 \times 84}{14} = 12 $, $ h_c = \frac{2 \times 84}{15} = 11.2 $. The shortest altitude is $ \boxed{11.2} $ km.

📌 Question: A hydrologist models a groundwater reservoir as a hemisphere with radius $ 3x $ and compares it to a cylindrical aquifer with radius $ x $ and height $ 3x $. What is the ratio of the hemisphere's volume to the cylinder's volume?

📌 Solution: The volume of a hemisphere is $ \frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi 27x^3 = 18\pi x^3 $. The cylinder's volume is $ \pi x^2 \cdot 3x = 3\pi x^3 $. The ratio is $ \frac{18\pi x^3}{3\pi x^3} = \boxed{6} $.

📌 Question: A hydrologist observes a circular drainage basin with area $ 100\pi \, \text{km}^2 $. If the radius decreases by 2 km due to sedimentation, by how many square kilometers does the area decrease?

📌 Solution: Original radius $ r = \sqrt{\frac{100\pi}{\pi}} = 10 \, \text{km} $. New radius $ r' = 8 \, \text{km} $. Original area $ 100\pi $, new area $ \pi (8)^2 = 64\pi $. The decrease is $ 100\pi - 64\pi = \boxed{36\pi} \, \text{km}^2 $.

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