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Home » This is a telescoping series, where most terms cancel out: - AMAZONAWS

This is a telescoping series, where most terms cancel out: - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 T(7) = 147 - 14 + 5 = 138

📌 \boxed{138}

📌 \]Question: A geographer is analyzing the distribution of settlements within a circular region on a map. If the rectangle formed by the extreme points of the region has sides of 10 km and 24 km, what is the circumference of the circle that circumscribes the rectangle?

📌 Solution: The diagonal of the rectangle is the diameter of the circumscribed circle. Using the Pythagorean theorem, the diagonal $ d = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 $ km. The circumference $ C = \pi \times d = 26\pi $ km. Thus, the circumference is $ \boxed{26\pi} $.

📌 Question: An online student studying STEM subjects is tasked with determining the ratio of the area of the incircle to the area of a right triangle with hypotenuse $ z $ and inradius $ c $. What is this ratio?

📌 Solution: The area of the incircle is $ \pi c^2 $. For a right triangle, the semiperimeter $ s = \frac{a + b + z}{2} $, and the inradius $ c = \frac{a + b - z}{2} $. The area of the triangle is $ r \times s = c \times (z + c) $. Therefore, the ratio is $ \frac{\pi c^2}{c(z + c)} = \frac{\pi c}{z + c} $. The answer is $ \boxed{\dfrac{\pi c}{z + c}} $.

📌 Question: A robotics researcher designs a circular obstacle avoidance path for a robot. If the robot's shortest detour forms a triangle with sides of 7 units, 24 units, and 25 units, what is the radius of the circumscribed circle around its path?

📌 Solution: For a right triangle with legs 7 and 24, and hypotenuse 25, the hypotenuse is the diameter of the circumscribed circle. The radius $ r = \frac{25}{2} = 12.5 $ units. Thus, the radius is $ \boxed{12.5} $.

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