Two faucets can fill a tank in 1 hour and 20 minutes. The first faucet takes more than two hours longer to fill the same tank when functioning without the second tap. How long does it take to fill each one separately? Explain your solution
time needed by 2nd faucet alone --- t hrs
rate for 2nd faucet alone = 1/t
time needed by 1st faucet alone --- t+2 hrs
rate of 1st faucet along = 1/(t+2)
combined rate = 1/t + 1/(t+2)
= (t+2 + t)/(t(t+2) = (2t+2)/(t(t+2))
then 1/[ 2t+2)/(t(t+2)) ] = 4/3
t(t+2)/(2t+2) = 4/3
3t^2 + 6t = 8t + 8
3t^2 - 2t - 8 = 0
(t-2)(3t+4) = 0
t = 2 or t = a negative time
the 2nd faucet would take 2 hours alone
the 1st facucet would take 4 hours alone
Let's denote the time it takes for the first faucet to fill the tank when functioning without the second faucet as 'x' hours.
We are given that when both faucets are functioning together, they can fill the tank in 1 hour and 20 minutes, which is equivalent to 1.33 hours.
So, if we consider the rate at which the faucets fill the tank, the combined rate is 1 tank per 1.33 hours, or 1/1.33 tanks per hour.
Now, let's consider the rates separately. The first faucet's rate when functioning alone is 1/x tanks per hour, and the second faucet's rate is 1/1.33 tanks per hour.
To determine the individual rates, we can set up the equation:
1/x + 1/1.33 = 1/1.33
Simplifying this equation, we get:
1/x = 0
Since x cannot be equal to zero, we know that the first faucet takes more than two hours longer than 0 hours to fill the tank by itself. Therefore, we can conclude that x must be greater than 2.
Hence, it takes the first faucet more than two hours longer to fill the tank by itself, but we do not have enough information to determine the exact time it takes to fill the tank separately for each faucet.
To find the time it takes to fill the tank separately with each faucet, we can set up a system of equations based on the given information.
Let's assume that the first faucet takes x hours to fill the tank on its own, and the second faucet takes y hours to fill the tank on its own.
From the information given, we can form two equations:
Equation 1: Two faucets together can fill the tank in 1 hour and 20 minutes, which is equal to 1 + 20/60 = 1.33 hours.
Therefore, the combined rate of both faucets is 1 tank per 1.33 hours:
1/x + 1/y = 1.33 ---------(1)
Equation 2: The first faucet takes more than two hours longer to fill the tank on its own compared to when both faucets are functioning.
So, the difference in time for the first faucet to fill the tank is 2 hours + 1.33 hours (the time it takes to fill with both faucets):
x - (1.33) = x - 1.33 = 2 ------------(2)
Now, we can solve these two equations simultaneously to find the values of x and y.
From equation (2), we have:
x - 1.33 = 2
Simplifying, we get:
x = 2 + 1.33
x = 3.33 hours
Now, substituting the value of x into equation (1):
1/3.33 + 1/y = 1.33
Simplifying further:
1/y = 1.33 - 1/3.33
1/y = 0.75
Taking the reciprocal of both sides, we get:
y = 1/0.75
y = 1.33 hours
Therefore, it takes 3.33 hours for the first faucet to fill the tank on its own, while it takes 1.33 hours for the second faucet to fill the tank on its own.
To summarize:
First faucet: 3.33 hours
Second faucet: 1.33 hours