Question 220108: A lanscaper balances his daily projects between small landscape jobs and mowing lawns. He allows 30 minutes for per lawn and 90 minutes per small landscape job. He works at most ten hours per day. The landscaper earns 35 per lawn and 125 per small landscape job. He can not do more than three landscape jobs per day and get all his mowing done. Find a combination of lawns mowed and completed landscape jobs per day that will maximize his income.
This is linear programming i really dont get this problem please help
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of lawnk jobs
y = number of landscape jobs
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revenue = number of lawn jobs * 35 and number of landscape jobs * 125
r = revenue
r = 35x + 125y
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30 minutes per lawn and 90 minutes per landscape job
works at most 10 hours per day
30 minutes equals .5 hours and 90 minutes equals 1.5 hours
z = total hours worked per day = 10
.5*x + 1.5*y <= 10
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no more than 3 landscaping jobs per day
y <= 3
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your revenue equation is:
r = 35x + 125y
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this is the equation you want to maximize
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your constraint equations are:
.5*x + 1.5*y <= 10
y <= 3
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the first constraint says that he can't work more than 10 hours per day on all jobs.
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the second constraint says that he can't work more than 3 landscape jobs and get all his lawn jobs done.
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the first thing I noticed was that 3 lawn jobs earns $105 while one landscape job that takes the same time earns $125.
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this should indicate that the landscape job gets him more money per hour.
I suspect the answer is that 3 landscape jobs and the rest lawn jobs will maximize his revenue for the day.
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let's see how that works.
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first you graph your constraints.
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the second constraint can be graphed as is.
that equation is y <= 3
the first constraint needs to be modified as follows:
.5x + 1.5y <= 10
subtract .5x from both sides of the equation and divide both sides of the equation by 1.5 to get:
y <= (10 - .5*x)/1.5
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you need to graph:
y = 3
y = (10-.5x)/1.5
It is shown below:
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if you look at the graph, you'll see the following.
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the line y = 3 is the maximum number of landscape jobs that can be done.
since landscaping jobs are represented by y, that means that y has to be smaller than or equal to 3. that means the area of the graph on or under the line y = 3 is where your solution will lie.
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the line y = (10-.5x)/1.5 is the maximum number of hours that he can spend on either job. that line is the one slanting down to the right. since the number of hours he can spend on either job is less than or equal to 10, the solution must lie on or underneath this line as well.
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Click on the following hyperlink to see a picture of this graph. Explanations will follow below the picture.
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Picture of Constraint Equations and Area of Possible Solutions
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The area beneath the line y = 3 and beneath the line .5x + 1.5y = 10 and above the x-axis is the area of possible solutions.
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The rules for linear programming indicate that a maximum / minimum solution will be at the points of intersection of all lines that bound the possible solution.
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In the picture you can see that:
One of the intersection points is at (0,3)
Another of the intersection points is at (11,3)
Another of the intersection points is at (0,0)
Another of the intersection points is (20,0)
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We examine each of the intersection points in the Revenue Equation. The Revenue Equation is y = 35*x + 125*y.
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At the point (0,3), this equation becomes:
35*0 + 125*3 = 375
At the point (11,3), this equation becomes:
35*11 + 125*3 = 385 + 375 = 760
At the point (0,0), this equation becomes:
35*0 + 125*0 = 0
At the point (20,0), this equation becomes:
35*20 + 125*0 = 700
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Remember x = number of lawn jobs and y = number of landscape jobs.
The maximum revenue is generated when 11 lawn jobs are done and 3 landscape jobs are done.
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As I pointed out earlier, this was to be expected since 3 lawn jobs earned less than 1 landscape job so logically doing the maximum number of lawn jobs possible earned the most money.
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