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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 C = 2\pi r = 2\pi \cdot \frac{13}{2} = 13\pi

📌 Thus, the circumference is:

📌 \boxed{13\pi \text{ cm}}

📌 Compute \(\tan 75^\circ\) using angle addition formulas.

📌 \tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}

📌 Rationalize the denominator:

📌 \]Question: A weather balloon is modeled as a sphere with radius $ r $ meters, and it ascends such that its volume increases at a rate proportional to $ \sin \theta $, where $ \theta $ is the angle of elevation from the ground. If the volume increase rate is given by $ \frac{dV}{dt} = k \sin \theta $, and $ \theta = 60^\circ $, compute the rate of volume increase in terms of $ k $ and $ r $.

📌 Solution: The volume of a sphere is given by $ V = \frac{4}{3} \pi r^3 $. Differentiating with respect to time $ t $, we get:

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