A Tank is normally filled in 9 hours but takes four hours longer to fill because of a leak in its bottom. If the tank is full, the leak will empty it in ?

Options

A. A) 32.5 hrs
B. B) 29.25 hrs
C. C) 30.30 hrs
D. D) 31 hrs

Answer

To solve the problem, we need to determine how long it will take for the leak to empty the tank if it is full. Let's break down the solution step by step.

### Step 1: Understand the filling process
The tank can normally be filled in 9 hours. This means that the filling rate of the tank is:
\[ \text{Filling rate} = \frac{1 \text{ tank}}{9 \text{ hours}} = \frac{1}{9} \text{ tanks per hour} \]

### Step 2: Determine the effect of the leak
Due to the leak, it takes 4 hours longer to fill the tank. Therefore, the total time taken to fill the tank with the leak is:
\[ \text{Total time with leak} = 9 \text{ hours} + 4 \text{ hours} = 13 \text{ hours} \]
The effective filling rate with the leak is:
\[ \text{Effective filling rate} = \frac{1 \text{ tank}}{13 \text{ hours}} = \frac{1}{13} \text{ tanks per hour} \]

### Step 3: Set up the equation
The effective filling rate can be expressed as the filling rate minus the leak rate. Let \( L \) be the leak rate (in tanks per hour). Thus, we have:
\[ \text{Filling rate} - \text{Leak rate} = \text{Effective filling rate} \]
Substituting the known values:
\[ \frac{1}{9} - L = \frac{1}{13} \]

### Step 4: Solve for the leak rate \( L \)
To solve for \( L \), we can rearrange the equation:
\[ L = \frac{1}{9} - \frac{1}{13} \]
To subtract these fractions, we need a common denominator. The least common multiple of 9 and 13 is 117. Thus, we rewrite the fractions:
\[ L = \frac{13}{117} - \frac{9}{117} = \frac{4}{117} \]
So, the leak rate \( L \) is:
\[ L = \frac{4}{117} \text{ tanks per hour} \]

### Step 5: Determine how long it takes for the leak to empty the tank
If the leak can empty the tank at a rate of \( \frac{4}{117} \) tanks per hour, we can find the time taken to empty 1 tank:
\[ \text{Time to empty the tank} = \frac{1 \text{ tank}}{L} = \frac{1}{\frac{4}{117}} = \frac{117}{4} = 29.25 \text{ hours} \]

### Final Answer
The leak will empty the tank in **29.25 hours**.

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