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Home » Star Butterfly Magic! Scientists Just Discovered Its Mysterious Powers Your Guide Reveals! - Upplift

Star Butterfly Magic! Scientists Just Discovered Its Mysterious Powers Your Guide Reveals! - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Solution: Using $\tan(60^\circ) = \frac{\text{height}}{\text{shadow length}}$, we substitute $\tan(60^\circ) = \sqrt{3}$ and shadow length $5\,\text{cm}$. Solving for height: $\text{height} = 5 \cdot \sqrt{3}$. Thus, $\boxed{5\sqrt{3}}$.

📌 Question: An equilateral triangle has an area of $25\sqrt{3}\,\text{cm}^2$. If each side is increased by $4\,\text{cm}$, by how many square centimeters does the area increase?

📌 Solution: The area of an equilateral triangle is $\frac{\sqrt{3}}{4} s^2$. Solving for original side $s$: $\frac{\sqrt{3}}{4} s^2 = 25\sqrt{3} \Rightarrow s^2 = 100 \Rightarrow s = 10\,\text{cm}$. New side: $14\,\text{cm}$. New area: $\frac{\sqrt{3}}{4} \cdot 14^2 = 49\sqrt{3}$. Increase: $49\sqrt{3} - 25\sqrt{3} = 24\sqrt{3}$. Thus, $\boxed{24\sqrt{3}}$.

📌 Question: A historical astrolabe features a right triangle

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Multiply both sides by \(x^2 + y^2 + z^2\):
9c^2(x^2 + y^2 + z^2) - 9c^2 z^2 = (x^2 + y^2 + z^2)^2.
9c^2(x^2 + y^2) = (x^2 + y^2 + z^2)^2.
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ho = 3c \sin(\phi)\), this describes a **sphere** of radius \(rac{3c}...
ho = 2a \sin\phi \Rightarrow ext{sphere of radius } a ext{ centered...
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