Question

pipe A can fill a pool in 3 hours while pipe B can fill in 5 hours.if the drain is open it can empty the pool in 8.5 hours.leaving the two pipes and the drain open,how long will it take to fill the pool?

Answer 1

The rate that pipe A fills is P/3 where P is the volume of the pool. The rate that pipe B fills is P/5. The rate that the drain empties the pool (- filling) is P/8.5. The rate of filling if both pipes are used would be P/3 + P/5. The net rate of filling if the drain also was open would be P/3 + P/5 - P/8.5

To find the time to fill the pool, set the volume of the pool (P) = the rate of filling * t (in hours)

\[P = (\frac{P}{3}+\frac{P}{5}-\frac{P}{8.5}) *t\] Solve for t.

Answer 2

i am clueless on how to solve it using your equation.

Answer 3

To solve for t, divide both sides by the big fraction. That gives \[t=\frac{P}{(\frac{P}{3}+\frac{P}{5}-\frac{P}{8.5})}=\frac{1}{(\frac{1}{3}+\frac{1}{5}-\frac{1}{8.5})}\]We can carefully punch that into a calculator, making good use of the 1/x key, or we can simplify some more by multiplying top and bottom by (3*5*8.5)\[t=\frac{1*(3*5*8.5)}{(\frac{1}{3}+\frac{1}{5}-\frac{1}{8.5})(3*5*8.5)}=\frac{127.5}{(5*8.5+3*8.5-3*5)}=\frac{127.5}{53}\approx 2.4\]

2.4 hours to fill the pool if pipe A and B are used and the drain is left open.

Always good to do a sanity check. A pumps 1/3 of a pool volume per hour, B pumps 1/5 of a pool volume per hour. Together, they do 1/3+1/5 = 5/15+3/15=8/15 of a pool volume per hour. The drain empties 1/8.5, but 8.5 is close to 7.5, so call it 2/15 per hour, giving a net fill of 8/15-2/15=6/15 per hour. After 2 hours, 6/15+6/15=12/15 is done, leaving only 15/15-12/15=3/15 to go, which will be done in 1/2 hour, giving us a total time of 2.5 hours. We slightly overestimate the drain rate, so it makes sense that our estimate is for slightly more time. Estimate is close to answer, and we understand why it differs, so our answer is probably right.

Answer 4

Because the P's appear on both sides of the equation, they cancel out. We wrote everything in terms of fractions of the pool volume, so we don't need to know the actual size of the pool — the answer holds whether it is a 50 gallon kiddie wading pool or a 50,000 gallon tank at the zoo.