A fire hose can fill a certain tank with water during 1 hour. A 2nd fire hose can fill the same tank in half hour. A 3rd fire hose can fill the same tank in quarter hour.

find the time necessary to fill the same tank by the 3 fire hoses together

1/T = 1/1 + 2/1 + 4/1 = 12,

T = (1/12)h = 5 Min.

Correction:

!/T = 1/1 + 2/1 + 4/1 = 7,
T = (1/7) hr. = 8.57 Min.

To find the time necessary to fill the tank when all three fire hoses are used together, we need to determine the combined filling rate of the three fire hoses.

Let's start by finding the filling rates of each fire hose:

- The first fire hose can fill the tank in 1 hour. So, its filling rate is 1 tank per 1 hour, which can be written as 1 tank/hour.
- The second fire hose can fill the tank in half an hour. So, its filling rate is 1 tank per 0.5 hours, which can be written as 2 tanks/hour.
- The third fire hose can fill the tank in a quarter hour. So, its filling rate is 1 tank per 0.25 hours, which can be written as 4 tanks/hour.

Now, to find the combined filling rate of all three hoses, we can add up their individual filling rates:

1 tank/hour + 2 tanks/hour + 4 tanks/hour = 7 tanks/hour

Thus, when all three fire hoses are used together, they have a combined filling rate of 7 tanks per hour.

Finally, to find the time necessary to fill the tank when all hoses are used together, we can divide the tank's capacity by the combined filling rate:

Time = Tank capacity / Combined filling rate
Time = 1 tank / 7 tanks/hour
Time = 0.143 hours or approximately 8.57 minutes

Therefore, it would take approximately 8.57 minutes to fill the tank when all three fire hoses are used together.