An application of a rational function is T = (AB)/(A+B), which gives the time, T, it takes for two workers to complete a particular task where A & B represent the time it would take for each individual worker to complete the identical task.
Estimate how long it takes you to complete a task of your choice (house cleaning, mowing, etc.) in a given week. Suppose that Joe is slower than you at the given task and takes twice as long as you do. If you work together, how long would it take you to complete the task?
Include the type of job, the time it takes you and Joe individually to complete the job, and the calculations needed to show how long it would take to complete the job if you worked together. Include units with your answer.
Problems of this type are solvable by either of the following methods.
<< If it takes me 2 hours to paint a room and you 3 hours, ow long will it take to paint it together? >>
Method 1:
1--A can paint the house in 5 hours.
2--B can paint the house in 3 hours.
3--A's rate of painting is 1 house per A hours (5 hours) or 1/A (1/5) houses/hour.
4--B's rate of painting is 1 house per B hours (3 hours) or 1/B (1/3) houses/hour.
5--Their combined rate of painting is 1/A + 1/B (1/5 + 1/3) = (A+B)/AB (8/15) houses /hour.
6--Therefore, the time required for both of them to paint the 1 house is 1 house/(A+B)/AB houses/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.
Note - T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.
Method 2:
Consider the following diagram -
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A
1--Let c represent the area of the house to be painted.
2--Let A = the number of hours it takes A to paint the house.
3--Let B = the number of hours it takes B to paint the house.
4--A and B start painting at the same point but proceed in opposite directions around the house.
5--Eventually they meet in x hours, each having painted an area proportional to their individual painting rates.
6--A will have painted y square feet and B will have painted (c-y) square feet.
7--From the figure, A/c = x/y or Ay = cx.
8--Similarly, B/c = x/(c-y) or by = bc - cx.
9--From 7 & 8, y = cx/a = (bc - cx)/b from which x = AB/(A+B), one half of the harmonic mean of A and B.
I think this should give you enough of a clue as to how to solve your particular problem.
Three people version
It takes Alan and Carl 40 hours to paint a house, Bill and Carl 80 hours to paint the house, and Alan and Bill 60 hours to paint the house. How long, to the nearest minute, will it take each working alone to paint the house and how long will it take all three of them working together to paint the house?
1--The combined time of two efforts is derived from one half the harmonic mean of the two individual times or Tc = AB/(A + B), A and B being the individual times of each participant.
2--Therefore, we can write
AC/(A + C) = 40 or AC = 40A + 40C (a)
BC/(B + C) = 80 or BC = 80B + 80C (b)
AB/(A + B) = 60 or AB = 60A + 60B (c)
3--From (a) and (c), 40C/(C - 40) = 60B/(B - 60)
4--Cross multiplying, 40BC - 2400C = 60BC - 2400B or BC = 120(B - C)
5--Equating to (b) yields 120(B - C) = 80(B + C)
6--Expanding and simplifying, 40B = 200C or B = 5C
7--Substituting into (b), 5C^2 = 400C + 80C = 480C making 5C = 480 or C = 96.
8--Therefore, B = 480 and A = 68.571
9--The combined working time of three individual efforts is derived from Tc = ABC/(AB + AC + BC)
10--Therefore, the combined time for all three to paint the house is
Tc = 68.571(480)96/[(68.571x480) + (68.571x96) + 480x96)) = 36.923 hours = 36 hr - 55.377 min
To estimate how long it would take you to complete a task of your choice, let's say we choose mowing the lawn. Suppose it takes you 4 hours to mow the lawn in a given week.
Given that Joe is slower than you and takes twice as long, it would take Joe 8 hours to mow the lawn.
To find out how long it would take you to complete the task together, we can use the rational function T = (AB)/(A+B), where A represents the time it takes you and B represents the time it takes Joe.
So, plugging in the respective values, we have T = (4 * 8)/(4 + 8) = 32/12 = 8/3.
Therefore, if you and Joe work together, it would take you approximately 8/3 hours to mow the lawn.