39. Filling a hemispherical tank A hemispherical tank with a radius of 10 m is filled from an inflow pipe at a rate of \( 3 \mathrm{~m}^{3} / \mathrm{min} \) (see figure). How fast is the water level rising when the water level is 5 m from the bottom of the tank? (Hint: The volume of a cap of thickness \( h \) sliced from a sphere of radius \( r \) is \( \pi h^{2}(3 r-h) / 3 \).) Inflow \( 3 \mathrm{~m}^{3} / \mathrm{min} \) 10 m
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- Radius of the hemisphere, \( r = 10 \) m. - Inflow rate, \( \frac{dV}{dt} = 3 \, \text{m}^3/\text{min} \). - Water level height, \( h = 5 \) m. - Volume of a cap: \( V = \frac{\pi h^2 (3r - h)}{3} \). Show more…
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Filling a hemispherical tank A hemispherical tank with a radius of $10 \mathrm{m}$ is filled from an inflow pipe at a rate of $3 \mathrm{m}^{3} / \mathrm{min}$ (see figure). How fast is the water level rising when the water level is $5 \mathrm{m}$ from the bottom of the tank? (Hint: The volume of a cap of thickness $h$ sliced from a sphere of radius $r$ is $\left.\pi h^{2}(3 r-h) / 3 .\right)$
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