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Real-World Applicationshard
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A hollow torus (donut-shaped tank) with major radius R=5R = 5R=5 meters and minor radius r=1r = 1r=1 meter is oriented vertically. Water leaks out of a small hole of area a=0.01a = 0.01a=0.01 m2^22 at the bottom of the torus under the influence of gravity (g=10g = 10g=10 m/s2^22). By Torricelli's Law, the rate of change of water volume is fracdVdt=−asqrt2gh\\frac{dV}{dt} = -a \\sqrt{2gh}fracdVdt=−asqrt2gh, where hhh is the current height of the water level from the bottom hole. When the water level is exactly at the center of the torus (h=r=1h = r = 1h=r=1 meter), what is the rate of change of the water height fracdhdt\\frac{dh}{dt}fracdhdt in meters per minute?