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There is a leak at the bottom of a tank. This leak can empty the full tank in 8 hours. When the tank is full, a tap is opened into the tank which admits 6 liters per hour, and the tank is emptied in 12 hours. What is the capacity of the tank?
A.\[28.8\] liters
B.36 liters
C.144 liters
D.145 liters

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Last updated date: 20th Jun 2024
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Answer
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Hint: We are given a tank with a hole in which the water is coming and from which the water is leaving. We are required to find the capacity of the tank. To solve this question, we will calculate the inlet and the outlet speed of the tank, and based on that we will get our total capacity.

Complete step-by-step answer:
Let us assume that the rate with which the tank empties is \[x\].
We are given that a hole can empty the full tank in 8hours.
Thus, the total amount of water that will flow out of the tank will be \[ = 8 \times x\].
Now, this will also be the capacity of the tank as the leak is emptying the full tank.
So, the capacity of the tank \[ = 8 \times x\]
Now, we are given that a tap is opened into the tank which admits water at 6 liters per hour. This will cause the rate of outflowing of the water to decrease.
Thus, the new rate of the outflow with which the tank will empty \[ = x - 6\]
Also, we are given that after the tap is opened the hole takes 12 hours. to empty the full tank.
Hence, the new rate of outflow will be \[ = 12 \times \left( {x - 6} \right)\]
This is also the capacity of the tank as the leak is emptying the full tank.
 So, we will now equate the two capacities. On doing so, we get
\[8 \times x = 12 \times \left( {x - 6} \right)\]
Multiplying the terms, we get
\[\begin{array}{l}8x = 12x - 72\\ \Rightarrow 8x - 12x = - 72\end{array}\]
Subtracting the like terms, we get
\[ \Rightarrow - 4x = - 72\]
Dividing both sides by \[ - 4\], we get
\[\begin{array}{l} \Rightarrow \dfrac{{ - 4x}}{{ - 4}} = \dfrac{{ - 72}}{{ - 4}}\\ \Rightarrow x = 18\end{array}\]
Thus, we get the rate of outflow of the tank as 18 liters/hours.
So now the capacity of the tank will be \[ = 8 \times x = 8 \times 18 = 144\]
Thus, the full capacity of the tank is 144 Liters.
Hence, the correct answer is option C


Note: We can check our answer by substituting the value of \[x = 18\]in the equation \[8 \times x = 12 \times \left( {x - 6} \right)\]. If the equation will be satisfied so our solution will be correct otherwise wrong.
Substituting \[x = 18\] in the equation, we get
\[8 \times 18 = 12 \times \left( {18 - 6} \right)\]
Multiplying the terms, we get
\[\begin{array}{l} \Rightarrow 144 = 12 \times 12\\ \Rightarrow 144 = 144\end{array}\]
As our equation is satisfied so our solution is correct.