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Question 970472: Sierra Wood Products manufactures two high-quality products, rockers and bookshelf units. Its profit is $30 per rocker and $42 per
bookshelf unit. Next week's production will be constrained by two limited resources, labor and wood. The labor available next week
is expected to be at most 930 hours, and the amount of wood available is expected to be at most 2400 board feet. Each rocker requires
6 labor hours and 8 board feet of wood. Each bookshelf unit requires 3 labor hours and 12 board feet of wood. Find how many rockers
and bookshelf units should be produced next week to maximize Sierra's profit. Find the maximum profit.
Let x=rocket and y=bookshelf unit
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your objective function is 30x + 42y
that's your profit and you want to maximize that.
your constraint equations are:
6x + 3y <= 930
8x + 12y <= 2400
x >= 0
y >= 0
solve for y in the first two equations and you get:
y <= (930-6x)/3
y <= (2400-8x)/12
x >= 0
y >= 0
x >= 0 and y >= 0 are the x and y axes respectively.
some graphing software allows you to do inequality equations.
the particular one i use allows that.
it's at http://www.desmos.com/calculator
put thoese equations into this graphing software and you get:
the darkest shaded region is the region of feasibility.
the corners of that region are the max/min points.
a little trick to make that region easier to see is to reverse all inequalities.
you will then get a graph that looks like this:
it's much easier to spot the region of feasibility this way, but you have to remember to reverse your inequalities.
now you analyze the objective function at the corner points.
you will get:
profit = 30x + 42y:
when s = 0 and y = 200, profit = 8400
when x = 82.5 and y = 145, profit = 8565
when x = 0 and y = 0, profit = 0
when x = 155 and y = 0, profit = 4650
maximum profit is achieved when x = 82.5 and y = 145.
that's 82.5 rockers and 145 bookshelves.
since you can't make 82.5 rockers, then 82 rockers will be the answer.
83 rockers could exhcuast the constraints.
let's take a quick look.
at 82 rockers and 145 bookshelves:
labor = 6 * 82 + 3 * 145 = 927
board feet = 8 * 82 + 12 * 145 = 2396
at 83 rockers and 45 bookshelves:
labor = 6 * 83 + 3 * 145 = 933
board feet = 8 * 83 + 12 * 145 = 2404
at 83 rockers, the constraints are exceeded.
at 82 rockers, the constraints have not been exceeded.
founding down to 82 from 82.5 was the way to go.
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