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Home » Therefore, the number of valid 3-digit codes is $oxed216$. - American Beagle Club

Therefore, the number of valid 3-digit codes is $oxed216$. - American Beagle Club

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 The volume \( V \) of the original cone with base radius \( r \) and height \( h = 2r \) is:

📌 V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3

📌 The scaled cone has dimensions \( kr \) and \( 2kr \), so its volume \( V_k \) is:

📌 V_k = \frac{1}{3} \pi (kr)^2 (2kr) = \frac{1}{3} \pi k^2 r^2 \cdot 2kr = \frac{2}{3} \pi k^3 r^3

📌 The ratio of the original volume to the scaled volume is:

📌 \frac{V}{V_k} = \frac{\frac{2}{3} \pi r^3}{\frac{2}{3} \pi k^3 r^3} = \frac{1}{k^3}

📌 Thus, the ratio is \(\boxed{\frac{1}{k^3}}\).

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Start with the equation:
\cos 2\theta = \sin \theta
Use the double-angle identity for cosine:
\cos 2\theta = 1 - 2\sin^2\theta
- 2\sin^2\theta = \sin \theta
Rearrange into a quadratic in \(\sin \theta\):
2\sin^2\theta + \sin \theta - 1 = 0
Let \( x = \sin \theta \), so:
2x^2 + x - 1 = 0
x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)} = \frac{-1 \pm \sqrt{1 +...

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