Each of three pumps A, B, C can pump water either into, or out of, a swimming pool. Pump A can fill the pool in 6 hours, while pumpp B works twice as fast. Pump C however takes one third more time than pump A to fill the pool.
1. All three pumps together pumping in water will fill the pool in ____ hours.
2. With pump B pumping out water, and both pumps A and C pumping in, the pool wil empty in _____ hours.
1/6 + 1/3 + 1/(4/3*6) = 5/8
So, they will take 8/5 hr or 1 hr 36 min.
Now just figure
1/3 - (1/6 + 1/8) = 1/24
so it takes 24 hrs to empty the pool
The speed of a boat in still water is 20 mph. It travels 24 mi upstream and 24 mi downstream in a total time of 10 hr. what is the speed if the current?
To solve these problems, we need to find the rates at which each pump can fill or empty the pool, and then use those rates to calculate the time required for each situation.
Let's denote the rate of pump A as "a" (pool filled per hour), the rate of pump B as "b", and the rate of pump C as "c".
1. To find the rate of pump A, we divide the pool capacity by the time it takes to fill it: a = 1/6 (since it takes 6 hours to fill the pool).
Pump B works twice as fast as pump A, so its rate is: b = 2a = 2/6 = 1/3.
Pump C takes one third more time than pump A, which means it takes 1/3 longer. So its rate is: c = a - (1/3)a = (2/3)a = (2/3) * (1/6) = 2/18 = 1/9.
Now, to find the combined rate when all three pumps are pumping in water together, we simply add their individual rates: a + b + c = 1/6 + 1/3 + 1/9.
To simplify this, we find a common denominator of 18: 3/18 + 6/18 + 2/18 = 11/18.
Therefore, all three pumps together can fill the pool at a rate of 11/18. Now, we can find the time required:
Time = 1 / (Rate) = 1 / (11/18) = 18/11 hours ≈ 1.64 hours.
So, all three pumps together pumping water will fill the pool in approximately 1.64 hours.
2. When pump B is pumping out water, its rate is negative since it's removing water from the pool. So, its rate would be -1/3 (negative one-third).
Now, to find the combined rate when pumps A and C are pumping in water while pump B is pumping out water, we subtract pump B's rate from the sum of pumps A and C: (a + c) - b = (1/6 + 1/9) - 1/3.
Again, we find a common denominator of 18: 3/18 + 2/18 - 6/18 = -1/18.
Therefore, the combined rate when pumps A and C are pumping in water (and pump B is pumping out) is -1/18. Now, we can find the time required to empty the pool:
Time = 1 / (Rate) = 1 / (-1/18) = -18 hours.
Since time cannot be negative, it means that the pumps are not able to empty the pool.
Therefore, with pump B pumping out water, and both pumps A and C pumping in, the pool will not empty.