A tank has four emptying taps, all of equal efficiencies, but each fixed at th, th, th and th of the height of the tank respectively. Two filling taps, each of which individually can fill the tank in 200 minutes, are also connected to the tank. If the ratio of the efficiency of each emptying tap and the efficiency of each filling tap is 1 : 3, and all the six taps are opened simultaneously, in how much time will the empty tank be filled?
Question
A tank has four emptying taps, all of equal efficiencies, but each fixed at th, th, th and th of the height of the tank respectively. Two filling taps, each of which individually can fill the tank in 200 minutes, are also connected to the tank. If the ratio of the efficiency of each emptying tap and the efficiency of each filling tap is 1 : 3, and all the six taps are opened simultaneously, in how much time will the empty tank be filled?
Solution
Let's break down the problem step by step:
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The efficiency of each emptying tap is 1/3 of the efficiency of each filling tap. Since there are 4 emptying taps, the combined efficiency of all emptying taps is 4 * (1/3) = 4/3.
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The efficiency of each filling tap is 1, and there are 2 filling taps. So, the combined efficiency of all filling taps is 2 * 1 = 2.
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When all taps are opened simultaneously, the net efficiency is the efficiency of all filling taps minus the efficiency of all emptying taps. So, the net efficiency is 2 - 4/3 = 2/3.
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The time taken to fill the tank is the reciprocal of the net efficiency. So, the time taken is 1 / (2/3) = 3/2 = 1.5 hours or 90 minutes.
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