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Home » \( x = 240 / 6.1051 ≈ 39.32 \) - AMAZONAWS

\( x = 240 / 6.1051 ≈ 39.32 \) - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Constant speed: \( d = 60 \times 20 = 1200 \, \text{m} \).

📌 Deceleration phase: \( a = \frac{-60}{5} = -12 \, \text{m/s}^2 \); distance \( = 60 \times 5 + \frac{1}{2}(-12)(5^2) = 300 - 150 = 150 \, \text{m} \).

📌 Total distance \( = 300 + 1200 + 150 = 1650 \, \text{m} \).

📌 A cone with height 12 cm and base radius 5 cm is filled with water. The water flows out through a small hole at the bottom at a rate proportional to the square root of the water height. If the outflow rate is \( k\sqrt{h} \), where \( k = 0.2 \, \text{cm}^{1.5}/\text{min} \), what is the height of water after 10 minutes?

📌 Volume of cone: \( V = \frac{1}{3}\pi r^2 h \); by similar triangles, \( r = \frac{5}{12}h \).

📌 So \( V = \frac{1}{3}\pi \left(\frac{5}{12}h\right)^2 h = \frac{25\pi}{432} h^3 \).

📌 Rate of change: \( \frac{dV}{dt} = -0.2\sqrt{h} \); but \( \frac{dV}{dt} = \frac{dV}{dh} \cdot \frac{dh}{dt} = \frac{25\pi}{432} \cdot 3h^2 \frac{dh}{dt} = \frac{25\pi}{144} h^2 \frac{dh}{dt} \).

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