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Home » Using \(\lim_u \to 0 \frac\sin uu = 1\), rewrite: - Groen Casting

Using \(\lim_u \to 0 \frac\sin uu = 1\), rewrite: - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Therefore, the ratio is \(\boxed{4}\).

📌 Question:** A food scientist is designing a conical container with height \( h \) and base radius \( r \), where \( h = 2r \). If they also consider a similar conical model with dimensions scaled by a factor of \( k \), what is the ratio of the volume of the original cone to the scaled cone?

📌 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}

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