Maple Rocker
Ozark Furniture Company can obtain at most 3000 feet of maple lumber for making its classic and modern maple rocking chairs. A classic maple rocker requires 15 board feet of maple, and a modern rocker requires 12 board feet of maple. Write an inequality that limits the possible number of maple rockers of each type that can be made, and graph the inequality in the first quadrant.
o Assign a variable to each type of rocker Ozark Furniture makes.
o Write a linear inequality which incorporates the given information of total board feet and the board feet required for each type of rocker.
o Draw a graph of the inequality so that you have this visual to go by as you discuss the graph in your writing.
o Format your math work as shown in the Instructor Guidance and be concise in your reasoning.
o Please make sure to include:
o Your solution to the above problem, making sure to include all mathematical work with an explanation as to what functions you performed.
o A discussion on what this graph looks like. Include information about the intercepts, the type of line needed, direction of the line, and region(s) shaded to fulfill the inequality. Any details which are pertinent to know about the graph should be mentioned.
o An application of the findings in this graph. For example, pick a point in the shaded area and give its coordinates, and then discuss what those numbers mean in terms of rockers and board feet of lumber. Pick a point outside of the shaded area and do the same thing. Pick a point right on the line and discuss the same details. Be specific.
o An evaluation of the following: A chain furniture store faxes an order for 175 modern rocking chairs and 125 classic rocking chairs. Will Ozark Furniture be able to fill this order with the current lumber on hand? If yes, how much lumber will they have left? If no, how much more lumber would they need to fill the order?
Answer
Let be the number of classic maple rockers any the number of modern maple rockers.
The quantity(in feet) of maple lumber required for making classic maple rockers and modern maple rockers is . The quantity of maple lumber available is 3000. Therefore the possible number of maple rockers of each type that can be made is limited by the inequality .
The possible number of maple rockers of each type is represented by the shaded region in the above diagram. The boundaries of the shaded region are the following three lines
the line , or the x-axis
the line , or the y-axis and
the line
The vertices of the triangular shaded region are the points
(1)the point or the origin which represents and . Making no classic maple rockers and no modern maple rockers. This point is the intersection of the x-axis and the y-axis.
(2)The point (200,0) or and . Making 200 classic maple rockers and no modern maple rockers. This point is the intersection of the x axis and the line
(3)The point (0,250) or and . Making no classic maple rockers and 250 modern maple rockers. This point is the intersection of the y axis and the line
A point in the shaded region
is a point in the shaded region. This represents and . Making 100 classic maple rockers and 100 modern maple rockers. The quantity of maple lumber required is
feet. The quantity of lumber left after making 100 classic maple rockers and 100 modern maple rockers = 3000-2700=300 feet.
A point outside the shaded region
is a point outside the shaded region. This represents and . Making 150 classic maple rockers and 100 modern maple rockers. The quantity of maple lumber required is
feet. The quantity of lumber available is only 3000 feet. Therefore this combination is not possible to make without additional lumber. The additional lumber required is 3750-3000=750 feet.
A point on the line
is a point in the shaded region. This represents and . Making 120 classic maple rockers and 100 modern maple rockers. The quantity of maple lumber required is
feet. The quantity of lumber available is also 300 feet. Therefore the quantity of lumber available is exactly that required for making this combination.
The order of 175 modern rocking and 125 classic rocking chairs.
and
The point (125,175) is outside the shaded region.
Quantity of lumber required = feet.
Quantity of lumber available = 3000 feet.
This combination is impossible to make without additional lumber. The additional lumber required is 3975-3000 = 975 feet.