Problem Analysis: Filling the Oil Tank
The problem asks for the total time to fill an oil tank. Initially, one tap fills the tank. After half the tank is filled, the number of taps increases. We need to calculate the time taken for both phases.
Step-by-Step Calculation
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Time for the First Half:
A single tap fills the entire tank in 14 hours. Therefore, the time taken by one tap to fill half ($\frac{1}{2}$) the tank is:
Time = Total Time $\times$ Fraction of Work = $14 \text{ hours} \times \frac{1}{2} = 7 \text{ hours}$.
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Rate of One Tap:
The rate of work for one tap is the amount of tank it can fill per hour:
Rate$_1 = \frac{1 \text{ tank}}{14 \text{ hours}} = \frac{1}{14}$ tank/hour.
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Taps for the Second Half:
After the first half is filled (which took 7 hours), 7 more similar taps are opened. The total number of taps working together is:
Total Taps = 1 (initial) + 7 (additional) = 8 taps.
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Combined Rate of Multiple Taps:
The combined rate of 8 taps is:
Combined Rate$_8 = 8 \times \text{Rate}_1 = 8 \times \frac{1}{14} = \frac{8}{14} = \frac{4}{7}$ tank/hour.
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Time for the Remaining Half:
The remaining work is to fill the other half ($\frac{1}{2}$) of the tank. The time required for 8 taps is:
Time$_2 = \frac{\text{Remaining Work}}{\text{Combined Rate}_8} = \frac{1/2 \text{ tank}}{4/7 \text{ tank/hour}} = \frac{1}{2} \times \frac{7}{4} \text{ hours} = \frac{7}{8} \text{ hours}.
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Convert Time to Hours, Minutes, and Seconds:
To convert $\frac{7}{8}$ hours to minutes and seconds:
- Minutes: $\frac{7}{8} \times 60 \text{ minutes} = \frac{420}{8} = 52.5 \text{ minutes}$.
- This is 52 minutes and 0.5 minutes.
- Seconds: $0.5 \times 60 \text{ seconds} = 30 \text{ seconds}$.
So, the time taken for the second half is 52 minutes and 30 seconds.
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Total Time Calculation:
The total time is the sum of the time taken for the first half and the second half:
Total Time = Time$_1$ + Time$_2$ = 7 hours + 52 minutes 30 seconds.
Total Time = 7 hours 52 minutes 30 seconds.
Final Answer
The total time taken to fill the oil tank completely is 7 hours 52 minutes 30 seconds.
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