|
Question 991871: Can you help me solve this?
A rectangular pool can be filled by two pipes together in 6 hours. The smaller pipe alone takes 5 hours longer than the larger pipe to fill the pool. Find the time each pipe will take to fill the pool on its own.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x is the rate of the larger pipe.
y is the rate of the smaller pipe.
when they work together, the equation becomes:
6x + 6y = 1
that means that the rate of the first pipe per hour times 6 hours plus the rate of the second pipe per hour times 6 hours equals 1 full pool.
when it works alone, the larger pipe takes z hours.
when it works alone, the smaller pipe take z+5 hours.
this is because the smaller pipe takes 5 more hours than the larger pipe to fill the pool on its own.
you get xz = 1 and you get y(z+5) = 1
xz = 1 means the rate of the larger pipe times z hours equals 1 full pool.
y(z+5) = 1 means the rate of the smaller pipe times z+5 hours equals 1 full pool.
solve the first of these equations for x and the second of these equations for y.
you will get:
x = 1/z
y = 1/(z+5)
you can now express x and y in terms of z.
go back to your original equation of 6x + 6y = 1
replace x with 1/z and y with 1/(z+5) to get:
6 * 1/z + 6 * 1/(z+5) = 1
multiply both sides of this equation by z * (z+5) to remove the denominators and you will get:
6 * (z+5) + 6 * z = 1 * z * (z+5)
simplify to get:
6z + 30 + 6z = z^2 + 5z
combine like terms to get:
12z + 30 = z^2 + 5z
subtract 12z from both sides of this equation and subtract 30 from both sides of this equation to get:
0 = z^2 5z - 12z - 30.
simplify further to get:
0 = z^2 - 7z - 30.
factor this equation to get:
0 = (z-10) * (z+3)
solve for z to get z = 10 and z = -3.
z can't be negative so you get z = 10.
since x = 1/z, you get x = 1/10.
since y = 1/(z+5), you get y = 1/15.
you now know the rates for the larger pipe and the smaller pipe and you also know how long it takes the larger pipe to fill the pool on its own and you also know how long it takes the smaller pipe to fill the pool on is own.
the rate of the larger pipe is 1/10 of the pool per hour.
the rate of the smaller pipe is 1/15 of the pool per hour.
go back to the original equation and replace x with 1/10 and y with 1/15 to get:
6x + 6y = 1 becomes 6/10 + 6/15 = 1
6/10 is equal to 3/5.
6/15 is equal to 2/5
3/5 + 2/5 is equal to 1
you get 1 = 1
this confirms the rates are good.
now go back to the original equations for the individual pipes to fill the pool.
the larger pipe fills the pool at the rate of 1/10 of the pool per hour.
that means it takes the larger pipe 10 hours to fill the pool on its own.
1/10 * z = 1
solve for z to get z = 10.
the smaller pipe fills the pool at the rate of 1/15 of the pool per hour.
that means it takes the smaller pipe 15 hours to fill the pool on its own.
1/15 * z = 1
solve for z to get z = 15.
the key to solving this using the method that i chose (there may be another method but i'm not aware of it) is to find the values of x and y in terms of z and then replacing x and y in the original equation with their equivalent values in terms of z.
this allows you to solve for z and, from there, to solve for x and y.
the variables of x, y and z are arbitrary.
you can use any variable name as long as you are consistent with what they represent.
|
|
|
| |
