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Home » Vibrant Valentines Coloring Sheets – Perfection for Fans of Love & Art! - ClickBalance

Vibrant Valentines Coloring Sheets – Perfection for Fans of Love & Art! - ClickBalance

📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 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 \( \frac{dV}{dt} = \frac{dV}{dh} \cdot \frac{dh}{dt} = \frac{25\pi}{432} \cdot 3h^2 \frac{dh}{dt} = \frac{25\pi}{144} h^2 \frac{dh}{dt} \).

📌 Thus, \( \frac{25\pi}{144} h^2 \frac{dh}{dt} = -0.2 \sqrt{h} \) → \( \frac{dh}{dt} = -\frac{0.2 \cdot 144}{25\pi} \cdot h^{-3/2} = -\frac{28.8}{25\pi} h^{-3/2} \).

📌 Separate variables and integrate from \( h = 12 \) at \( t = 0 \) to \( t = 10 \):

📌 \( \int_{12}^{h} h^{3/2} dh = -\frac{28.8}{25\pi} \int_0^{10} dt \)

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