Question

it takes 24 hours to fill a swimming pool using two pipes. if the pipe of larger diameter is used for 8 hours and the pipe of the smaller diameter is used for 18 hours . only half of the pool is filled. how long would each pipe take to fill the swimming pool.?

Answer 1

Answer:

Larger pipe takes 40 hours and smaller pipe takes 60 hours

Step-by-step explanation:

Define x and y:

Let x be the number of hours needed for the larger pipe to fill the tank.

1 hour = 1/x of the tank

Let y be the number of hours needed for the smaller pipe to fill the tank.

1 hour = 1/y of the tank

Find the capacity of the tank in term of x and y:

It takes 24 hour to fill the tank if both pipes are used

1 hour = 1/x + 1/y = (x + y)/xy

Time needed to fill the tank = xy/(x + y)

Form equation:

Time needed to fill the tank = 24 hours

xy/(x + y) = 24

xy = 24(x + y)

xy = 24x + 24y

xy - 24y = 24x

y(x - 24) = 24x

y = 24x/(x - 24)

Find the amount of pool filled with the larger pipe is used for 8 hours

1 hour = 1/x

8 hours = 8/x

Find the amount of pool filled with the smaller pipe is used for 18 hours

1 hour = 1/y

18 hours = 18/y

Form equation:

It fills half the tank:

8/x + 18/y = 1/2

(8y + 18x)/xy = 1/2

2(8y + 18x) = xy

16y + 36x = xy

xy - 16y = 36x

y(x - 16) = 36x

y = 36x/(x - 16)

Solve x and y:

y = 24x/(x - 24)    ------------------- [ 1 ]

y = 36x/(x - 16)       ------------------- [ 2 ]

Equation [ 1 ] = [ 2 ]

24x/(x - 24) =  36x/(x - 16)

24x(x - 16) = 36x(x - 24)

24x² - 384x = 36x² -  864x

12x² - 480x = 0

x² - 40x = 0

x(x - 40) = 0

x = 0 or x = 40

Find y:

y = 24x/(x - 24)

y = 24(40)/(40 - 24)

y = 60

Find the number of hours needed for each pipe:

Larger pipe = x = 40 hours

Smaller pipe = y = 60 hours

Answer: Larger pipe takes 40 hours and smaller pipe takes 60 hours