Question

Remember the Peach Orchard at the Amusement Park? You are trying to conserve water, including the water used to irrigate the orchards. Currently, you are using two sprinkler systems, and you need to analyze these systems. Together, the two systems water in 1.5 hours. Sprinkler System B by itself takes 4 hours longer than Sprinkler System A by itself. How long does each system take to water the orchard alone? Now that you have the rates of each individual system, we need to determine the combined rate. The two systems together take 1.5 hours to water the orchard, so the rate is: 2/3 orchards/hour Use this, combined with your responses to questions 1, 2, and 3, to create an equation that represents the relation of the three rates, then solve for the variable t to find the time that Sprinkler System A takes to water the orchard. How long does it take Sprinkler System A and Sprinkler System B to water the orchard alone?

Answer 1

We are given that:

t = time required for Sprinkler System A

t + 4 = time required for Sprinkler System B

 

Hence the rates are:

1 / t = rate for Sprinkler System A

1 / (t + 4) = rate for Sprinkler System B

 

The overall equation is:

1 / t  + 1 / (t + 4) = 1 / 1.5

 

Solving for t by multiplying everything by t*(t + 4):

t + 4 + t = (1 / 1.5) (t) (t + 4)

3 t + 6 = t^2 + 4 t

t^2 + t = 6

Completing the square:

t^2 + t + 0.25 = 6 + 0.25

(t + 0.5)^2 = 6.25

t = -0.5 ± 2.5

t = -3, 2

 

Since time cannot be negative, therefore:

t = 2 hours

t + 4 = 6 hours

 

Alone, Sprinkler System A takes 2 hours while Sprinkler System B takes 6 hours.