Pipes A and B can fill a storage tank in 8 hours and 12 hours, respectively. With the tank empty, pipe A was turned on at noon, and then pipe B was turned on at 1:30 P.M. At what time was the tank full?
A does (1/8) tank/hr
B does (1/12) tank/hr
how long to fill
(1/8)t + (1/12)(t-1.5) = 1 tank full
3 t + 2t - 3 = 24
5 t = 27
t = 27/5 = 5 2/5 hr
2/5 * 60 = 24 min
so 5:24 pm
To solve this problem, we need to find the rate at which each pipe fills the tank and then calculate the combined rate when both pipes are turned on. Let's start by finding the rate at which each pipe fills the tank:
Pipe A fills the tank in 8 hours, so its rate is 1/8 of the tank per hour.
Pipe B fills the tank in 12 hours, so its rate is 1/12 of the tank per hour.
Now, let's calculate the combined rate:
When pipe A is turned on at noon, it has been running for a total of 1 hour until 1 P.M.
During this time, pipe A fills 1/8 of the tank.
At 1:30 P.M., pipe B is turned on, which means it has been running for a total of 0.5 hours until 2 P.M.
During this time, pipe B fills 0.5/12 = 1/24 of the tank.
From 2 P.M., both pipes A and B are running simultaneously. Therefore, their combined rate is:
1/8 + 1/24 = 3/24 + 1/24 = 4/24 = 1/6 of the tank per hour.
Now, we know that the tank is empty at noon and we need to find the time it takes for the combined rate to fill the entire tank, which is 1.
Since the combined rate is 1/6 of the tank per hour, it will take 6 hours to fill the entire tank (1 / 1/6 = 6).
Adding the 6 hours to 2 P.M., we find that the tank will be full at 8 P.M.
Therefore, the tank will be full at 8 P.M.
To determine the time it takes to fill the tank when both pipes A and B are working together, we need to find their combined fill rate.
Let's assign a variable to represent the fill rate of pipe A per hour. Since pipe A can fill the tank in 8 hours, its fill rate would be 1/8 of the tank per hour (1 tank / 8 hours = 1/8 tank per hour).
Similarly, let's assign another variable for the fill rate of pipe B per hour. Since pipe B can fill the tank in 12 hours, its fill rate would be 1/12 of the tank per hour (1 tank / 12 hours = 1/12 tank per hour).
When both pipes A and B are working together, their combined fill rate will be the sum of their fill rates: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 tank per hour.
Now, we know that pipe A started at noon, which means it worked for 1.5 hours until pipe B was turned on at 1:30 P.M. Pipe B, on the other hand, only worked for the remaining time until the tank is full. Let's use T to represent the time pipe B worked.
Therefore, the equation representing the filling of the tank is:
1.5 * (1/8) + T * (5/24) = 1
Let's solve for T:
1.5/8 + T * 5/24 = 1
Multiply both sides by 24 to get rid of the fractions:
3 + 5T = 24
Subtract 3 from both sides:
5T = 21
Divide both sides by 5:
T = 21/5
Now we can convert the fraction to a mixed number:
T = 4 1/5
Therefore, pipe B worked for a total of 4 hours and 12 minutes (1/5 of an hour).
To find the time when the tank was full, we add the time that pipe A worked (1.5 hours) to the time that pipe B worked (4 hours and 12 minutes):
1.5 + 4 hours + 12 minutes = 5.5 hours + 0.2 hours = 5.7 hours
So, the tank was full after approximately 5 hours and 42 minutes.
Adding this time to the starting time of pipe A (noon), we can determine that the tank was full at:
12:00 PM + 5 hours and 42 minutes = 5:42 PM