A water tank can be filled up by two pipes together in 30 minutes. One pipe fills up the tank in 32 min sooner than the other pipe when they are used separately. Find the time for each pipe to fill up the tank.
Can you explain how you set up the equation thx
As with all such problems, consider how much of the job is done by each worker:
1/x + 1/(x-32) = 1/30
x=80,12
12 is bogus, since that would make x-32 negative.
So, the pipes can fill the tank in 80 and 48 minutes
To set up the equation, let's assume that the slower pipe takes x minutes to fill up the tank on its own. This means that the faster pipe takes (x - 32) minutes to fill up the tank on its own, as stated in the question.
When both pipes are operating together, they can fill up the tank in 30 minutes. Therefore, their combined rate of filling the tank is 1/30 of the tank per minute.
Now, let's consider the rates at which each pipe fills up the tank individually. The slower pipe fills up 1/x of the tank per minute, while the faster pipe fills up 1/(x - 32) of the tank per minute.
To solve the equation, we can add up the rates of both pipes to get the combined rate:
1/x + 1/(x - 32) = 1/30
To solve this equation, we can multiply through by the common denominator, which is 30x(x - 32), to simplify:
30(x - 32) + 30x = x(x - 32)
Now, we can expand and simplify the equation:
30x - 960 + 30x = x^2 - 32x
Combining like terms:
60x - 960 = x^2 - 32x
Next, move all the terms to one side to set the equation equal to zero:
x^2 - 92x + 960 = 0
Now, we have a quadratic equation, which we can solve using factoring, completing the square, or the quadratic formula.
By factoring:
(x - 40)(x - 24) = 0
Therefore, x can be either 40 or 24. Recall that x represents the time taken by the slower pipe to fill up the tank on its own.
Thus, the time taken by the slower pipe is 40 minutes, and the time taken by the faster pipe is (40 - 32) = 8 minutes.