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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: A cartographer uses satellite data to map a triangular atoll with side lengths 13 km, 14 km, and 15 km. To optimize GPS signal reception, a circular transmission range must cover the entire triangle. Find the radius of the circumcircle of this triangle, express your answer in simplified radical form.

📌 Solution: We are given a triangle with sides $a = 13$, $b = 14$, and $c = 15$. The circumradius $R$ of a triangle with sides $a$, $b$, $c$ is given by the formula:

📌 where $K$ is the area of the triangle. First, compute the semi-perimeter $s$:

📌 Now use Herons formula to find the area $K$:

📌 Solution: Let the side length of the original equilateral triangle be $s$. The area $A$ of an equilateral triangle is:

📌 After reducing each side by 4 mm, the new side length is $s = 12$ mm. The new area is:

📌 Question: An oceanographer analyzing spherical underwater sensor buoy designs compares a full buoy of radius $2r$ to a hemispherical float of radius $r$. What is the ratio of the volume of the buoy to the volume of the float?

📌 Solution: Volume of a sphere with radius $2r$ is:

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