5 (12 points). Torricelli's principle says that the speed at which fluid exits a punc- tured...

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5 (12 points). Torricelli's principle says that the speed at which fluid exits a punc- tured vessel is proportional to the square root of the distance between the surface and the hole. Suppose a hemispherical bowl is filled to the brim with 21 L of Five Alive and then pierced at the bottom by a 1 cm thick rusty nail. Let V and Y be the volume and depth, respectively, of the juice in the bowl at time t. (a) Find a formula for dV/dy using slices. (b) Find a formula for dv/dt using Torricelli's principle. (c) Using (a) and (b), write a first-order differential equation for y in terms of t. -1 (d) Assuming the constant of proportionality is 4.4 m¹/2s-¹, when will the juice run out? Round your answer to the nearest minute. 5 (12 points). Torricelli's principle says that the speed at which fluid exits a punc- tured vessel is proportional to the square root of the distance between the surface and the hole. Suppose a hemispherical bowl is filled to the brim with 21 L of Five Alive and then pierced at the bottom by a 1 cm thick rusty nail. Let V and Y be the volume and depth, respectively, of the juice in the bowl at time t. (a) Find a formula for dV/dy using slices. (b) Find a formula for dv/dt using Torricelli's principle. (c) Using (a) and (b), write a first-order differential equation for y in terms of t. -1 (d) Assuming the constant of proportionality is 4.4 m¹/2s-¹, when will the juice run out? Round your answer to the nearest minute.

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