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Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then
9. The work required to empty the top half of the tank
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- The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…arrow_forwardThere is a tank in the shape of a paraboloid: Water is being pumped into the tank at a constant rate of 2π cubic inches per minute. If y denotes the water level, then the volume of water is given by V = πy2 / 2. (a) Suppose that the height of the paraboloid is 10 inches. Let a denote the rate of change of water level when y = 1 inch and let b denote the rate of change of water level when y = 5 inches. Without computing anything, do you suspect that a or b is greater? (b) Confirm your suspicion by explicitly computing a and barrow_forward
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