Final answer:
The total time taken to fill the tank considering the rates of the two pipes and the leak, the effective rate of filling the tank is calculated. It takes 6 additional hours after the leak starts, plus 1 hour for the initial filling, totaling 7 hours to fill the tank.
Step-by-step explanation:
The student asked about the total time taken to fill a tank with two pipes filling it in 10 hours and 15 hours separately and a leak developing when the tank is 1/6 full, causing 1/6 of the water to leak out. To solve this question, we must consider the rate at which the pipes can fill the tank and then adjust that rate once the leak begins.
Firstly, we find the rate of each pipe: Pipe A fills the tank in 10 hours, so its rate is 1/10 of the tank per hour; Pipe B fills the tank in 15 hours, so its rate is 1/15 of the tank per hour. When both pipes are opened together, they fill at a combined rate of (1/10 + 1/15) tanks per hour, which simplifies to 1/6 tanks per hour. The tank is filled to 1/6 of its capacity when the leak starts, so the remaining capacity to be filled is 5/6.
Now, we incorporate the leak, which wastes 1/6 of the water coming in. This means that only 5/6 of the water from both pipes actually contributes to filling the tank. Therefore, the effective rate of filling is 5/6 of the combined rate, which is 5/6 * 1/6 = 5/36 tanks per hour. We then divide the remaining capacity (5/6 tanks) by this effective rate (5/36 tanks per hour) to find out how many more hours it takes to fill the tank, which is (5/6) / (5/36) = 6 hours. Adding the time taken to fill the initial 1/6 of the tank, which is 1 hour (since the tank fills in 6 hours, each 1/6 takes 1 hour), the total time taken to fill the tank is 7 hours.