The first of three pipes fills a tank in 12 minutes and a second pipe in 10 minutes.If both these pipes and also a third pipe are open, the tank is filled in 4 minutes. How many minute will the third pipe alone require to fill the tank?
Just need an equation.
Thank you!
Let tank volume = V in gallons
First pipe fill rate = Q1 = V/12 g.p.m
Second pipe fill rate = Q2 = V/10 g.p.m
Let Q3 = third pipe fill rate in gallons/min
4 minutes = V/(Q1 + Q2 + Q3)
= V/(V/12 + V/10 + Q3).
= 1/(1/12 + 1/10 + Q3/V)
1/3 + 2/5 + 4 Q3/V = 1
4 Q3/V = 4/15
Q3/V = 1/15
Fill time with pipe 3 alone = V/Q3 = 15 minutes
To solve this problem, let's assign variables to represent the rates at which each pipe can fill the tank.
Let's say:
- The rate at which the first pipe fills the tank is R1 (in tank per minute).
- The rate at which the second pipe fills the tank is R2 (in tank per minute).
- The rate at which the third pipe fills the tank is R3 (in tank per minute).
From the given information, we know that:
- The first pipe takes 12 minutes to fill the tank, so its rate is 1 tank per 12 minutes: R1 = 1/12 tank per minute.
- The second pipe takes 10 minutes to fill the tank, so its rate is 1 tank per 10 minutes: R2 = 1/10 tank per minute.
- When all three pipes are open, the tank is filled in 4 minutes, so the combined rate of all three pipes is 1 tank per 4 minutes: R1 + R2 + R3 = 1/4 tank per minute.
Now, we need to find the rate at which the third pipe fills the tank (R3) in order to determine how many minutes it would take for the third pipe alone to fill the tank.
Using the information above and the equation for the combined rate, we can calculate the rate of the third pipe:
R1 + R2 + R3 = 1/4
1/12 + 1/10 + R3 = 1/4
10/120 + 12/120 + R3 = 30/120
22/120 + R3 = 30/120
R3 = 30/120 - 22/120
R3 = 8/120
R3 = 1/15
Therefore, the rate at which the third pipe fills the tank is 1/15 tank per minute.
So, to find out how long the third pipe alone would take to fill the tank, we can set up the equation:
1 tank / X minutes = 1/15 tank per minute
Simplifying this equation, we can solve for X:
X = 15 minutes
Therefore, the third pipe alone would require 15 minutes to fill the tank.