Question 122845: I struggle with word problems and I am really struggling with these two. If someone could please help me understand these I would be very grateful. I have been working on these two problems for 2 days and I believe I am confusing myself due to the nature of the problem. Thanks!!
Question 1
A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.
Question 2
A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities.
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! Question 1
.A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.
:
Let x = standard p.h; y = deluxe p.h
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The labor equation:
12x + 20y =< 800
:
put equation in the general (y=) form to plot the graph
20y =< 800 - 12x
y =< 800/20 - (12/20)x
y =< 40 - .6x
:
:
The production capacity equation:
x + y =< 50
Put this in the "y=" form also
y <= 50 -x
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Existing order constraints
x => 10
and
y => 15
:
Plot these using the equation givens.
y = 40 - .6x; (purple line)
y = 50 - x; (green line)
y = 15; Note that y = 15 is a horizontal line going thru y = 15; black line
x = 10 is a vertical line going thru x = 10
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I assume you know how to make up an x/y table and plot a graph, for each of these equations. (I show a graphing procedure in the 2nd problem)
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Here is the graph:
:
That feasibility area is:
1.At or below, the purple line or the green line whichever is lower.
2.At or above the black horizontal line
3.At or to the right of the vertical line
:
:
Question 2
A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities.
:
x = no. of scarves; y = no. of sweaters:
:
The labor (in hours) inequality:
2x + 14y =< 40
Arrange in the general (y=) form so we can graph it:
14y =< -2x + 40
y =< -(2/14)x + (40/14)
y =< -(1/7)x + 20/7
:
Plot this equation:
For x = 0
y = -(1/7)(0) + 20/7
y = 20/7 or about 2.85
:
For x = 7
y = -(1/7)(7) + (20/7)
y = -1 + 20/7
y = -(7/7) + (20/7)
y = +13/7 or about 1.85
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A table of the x/y coordinates:
x | y
-------
0 | 2.85
7 | 1.85
Join these two points with a straight-edge for the labor graph
:
:
The production inequality:
x + y = 5
y = 5 - x
:
Plot this equation:
For x = 0, then y = 5, obviously
:
For x = 4
y = 5 - 4
y = 1
:
The table for these x/y coordinates
x | y
-------
0 | 5
4 | 1
Join these to points for the production graph
:
your graph should look like this:
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The area of feasibility would be at or below either line, whichever is lower
It's assumed that x and y => 0
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It looks like the best you could do is 3 scarfs and 2 sweaters
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