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Home » The volume \(V_\textsphere\) of a sphere with radius \(r\) is: - Crosslake

The volume \(V_\textsphere\) of a sphere with radius \(r\) is: - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A glaciologist models a glacier's movement using a right triangle with legs measuring 15 km and 20 km. If the shorter leg is increased by 5 km, by how many square kilometers does the area increase?

📌 Solution: Original area: \( \frac{1}{2} \times 15 \times 20 = 150 \) km². New shorter leg: \( 15 + 5 = 20 \) km. New area: \( \frac{1}{2} \times 20 \times 20 = 200 \) km². The increase is \( 200 - 150 = 50 \) km². \boxed{50}

📌 Question: Compute \( \sin 225^\circ $.

📌 Solution: \( 225^\circ \) lies in the third quadrant, where sine is negative. Reference angle: \( 225^\circ - 180^\circ = 45^\circ \). Thus, \( \sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2} \). \boxed{-\dfrac{\sqrt{2}}{2}}

📌 Question: A tidal energy system uses a rectangular frame with dimensions 8 meters by 15 meters. If each dimension is reduced by 3 meters, by how many square meters does the area decrease?

📌 Solution: Original area: \( 8 \times 15 = 120 \) m². New dimensions: \( 8 - 3 = 5 \) meters and \( 15 - 3 = 12 \) meters. New area: \( 5 \times 12 = 60 \) m². The decrease is \( 120 - 60 = 60 \) m². \boxed{60}Question: A geologist is studying the periodic patterns of seismic waves and notes that a specific wave pattern repeats every 24 seconds. If the current time is 2025-06-28 18:21:16, what is the smallest positive time (in seconds) after 18:21:16 when the wave pattern will repeat, such that this time is congruent to 6 (mod 11)?

📌 Solution: Let \( t \) be the time in seconds after 18:21:16 when the pattern repeats and satisfies \( t \equiv 6 \pmod{11} \). Since the wave pattern repeats every 24 seconds, we are looking for the smallest \( t \) such that:

📌 t = 24k \quad \text{and} \quad 24k \equiv 6 \pmod{11}

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