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5Question: A triangle has sides measuring 13 cm, 14 cm, and 15 cm. What is the length of the shortest altitude drawn to the longest side?
Question: A science teacher uses a right circular cone to model a volcano. The cone has a base radius of 5 cm and height of 12 cm. What is the slant height of the cone, and what is the angle between the slant height and the base?
Question: A science communicator creates a 3D animation of a hemisphere and a sphere for a video. If the hemisphere has a radius of $ 3x $ units and the sphere has a radius of $ x $ units, what is the ratio of the surface area of the hemisphere (including the base) to that of the sphere?
Let us define a substitution to simplify the expression. Set:
Question: Let $ x, y, z $ be positive real numbers. Find the minimum value of
Since $ x, y, z $ are positive real numbers, equality is achievable.
This is a functional equation. Lets analyze the condition:
So $ f $ satisfies the Cauchy functional equation:
Solution: We apply the AM-GM inequality: