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Question 349529: Beth has two part-time jobs: tutoring and law mowing. She wants to work NO MORE than 12 hours per week; at least 4 hours tutoring and not more than 6 hours mowing. She makes $5/hour tutoring and $6.50/hour mowing. What combination of hours would make her the most money?
Identify the constraints:
Graph this on an x-y axis
Write an objective function
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = hours spent tutoring.
let y = hours spend lawn mowing.
x + y < 12 is one equation.
x >= 4 is another equation.
y <= 6 is another equation.
y >= 0 is another equation (number of hours can't be negative).
All 4 are constraint equations.
The equation she is looking to maximize is the revenue equation.
That equation is R = 5*x + 6.5*yt
She needs to graph the constraint equations and then look at the intersection points to find the maximum revenue.
To graph the equations, make them an equality rather than an inequality.
you get:
x + y = 12 (E2 in the picture)
x = 4 (E1 in the picture)
y = 6 (E3 in the picture)
y = 0 (E4 in the picture)
x + y = 12 is equivalent to y = 12 - x when you solve for y.
replace x + y = 12 with y = 12 - x and your equations are:
y = 12 - x (E2 in the picture)
x = 4 (E1 in the picture)
y = 6 (E3 in the picture)
y = 0 (E4 in the picture)
your intersection points for these 4 constraint equations are:
(x,y) = :
(4,0)
(4,6)
(6,6)
(12,0)
Solve the revenue equation at these intersection points.
revenue equation is:
R = 5*x + 6.5*y
at (4,0) this equation becomes 5*4 + 6.5*0 = 20
at (4,6) this equation becomes 5*4 + 6.5*6 = 59
at (6,6) this equation becomes 5*6 + 6.5*6 = 69
at (12,0) this equation becomes 5*12 + 6.5*0 = 60
maximum revenue occurs at (6,6).
6 hours of tutoring and 6 hours of mowing.
it meets the constraints since hours of tutoring is greater than or equal to 4 and hours of mowing is less than or equal to 6 and total hours is less than or equal to 12.
obviously, if hours of mowing was allowed to be more than 6, she would spend more hours mowing because mowing gives her 6.5 per hours while tutoring only gives her 5, but the constraints forced hours of mowing to be equal to or less than 6.
note that the maximum hours allowed for mowing is the maximum revenue solution.
this won't always be the case, but it is here because there are no other constraints that would not make it so.
the picture of the constraint equations is shown below.
the shaded area is the area in which all constraint are met.
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