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the cast of the avatar the last airbender - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{15(15 - 7)(15 - 10)(15 - 13)} = \sqrt{15 \cdot 8 \cdot 5 \cdot 2}

📌 A = \sqrt{15 \cdot 8 \cdot 10} = \sqrt{1200} = \sqrt{400 \cdot 3} = 20\sqrt{3}

📌 r = \frac{A}{s} = \frac{20\sqrt{3}}{15} = \frac{4\sqrt{3}}{3}

📌 Thus, the radius of the inscribed circle is $ \boxed{\dfrac{4\sqrt{3}}{3}} $ cm.

📌 Question: An entomologist observes a spiderweb forming a perfect equilateral triangle with side length 12 cm. What is the area of the circle that circumscribes this triangle?

📌 For an equilateral triangle of side length $ s = 12 $, the circumradius $ R $ is given by the formula:

📌 R = \frac{s}{\sqrt{3}}

📌 But more accurately, the correct formula for the circumradius of an equilateral triangle is:

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4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115 \implies 4(x - 4)^2 - 64 -...
4(x - 4)^2 - 9(y - 3)^2 + 17 = -115 \implies 4(x - 4)^2 - 9(y - 3)^2 =...
Divide by $ -132 $ to standardize:
\frac{(y - 3)^2}{\frac{132}{9}} - \frac{(x - 4)^2}{\frac{132}{4}} = 1.
The center is at $ (h, k) = (4, 3) $.
Final answer: $\boxed{(4, 3)}$
Question: Microfossil density $ d(t) $ in a sediment layer follows $ \...
Solution: Use partial fractions to decompose $ \frac{1}{k(k+1)(k+2)} $...
\frac{1}{k(k+1)(k+2)} = \frac{A}{k} + \frac{B}{k+1} + \frac{C}{k+2}.
Multiply through: $ 1 = A(k+1)(k+2) + Bk(k+2) + Ck(k+1) $.

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