Final answer:
To determine how long three pipes with different diameters would take to fill a cistern together, we calculate the rate of each based on the square of their diameters and the fact that the largest pipe fills the cistern in 58 minutes. We then add these rates to find the combined rate and take the reciprocal to find the time required to fill the cistern with all pipes running together.
Step-by-step explanation:
The question asks how long it would take for three pipes with diameters of 2 cm, 3 cm, and 4 cm to fill a cistern if they were all running together, given that the largest pipe (4 cm in diameter) can fill the cistern alone in 58 minutes. Since the flow of water is proportional to the square of the diameter, we can determine the rate at which each pipe fills the cistern separately, then find the combined rate when all are used together.
We denote the rate of the largest pipe as R. The other pipes have diameters that are 1/2 (2 cm) and 3/4 (3 cm) of the largest pipe's diameter. The rate of flow through a pipe (which determines how quickly it can fill the cistern) is proportional to the square of its diameter, so it follows that the rates of flow for the three pipes are proportional to the squares of 1/2, 3/4, and 1, respectively. This gives us the rates as 1/4R, 9/16R, and R for the pipes with diameters 2 cm, 3 cm, and 4 cm, respectively.
Calculating the total rate: If the largest pipe has a rate R and takes 58 minutes to fill the cistern, then R = 1/58 cisterns/minute. The sum of the rates for all three pipes is 1/4R + 9/16R + R, which gives us 1/4(1/58) + 9/16(1/58) + 1(1/58). We can solve this to find the combined rate of the three pipes.
Finding the time: To determine the total time it would take for all three pipes to fill the cistern together, we would take the reciprocal of the total rate as found above. This gives us the time in minutes for the cistern to be filled with all three pipes running.