Question) Maple rockers. Ozark Furniture Company can obtain at most 3000 board 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.
Solution) As per the question statement,
Total Maple Lumber available= 3000 Board feet
Maple Lumber required for a classic maple rocker= 15 board feet
Maple Lumber required for a modern rocker= 12 board feet
Let, the number of classic maple rockers and modern rockers which can be made out of the available quantity of maple lumber be x and y respectively.
Therefore, Quantity of maple lumber required for x classic maple rockers= 15* x board feet
Quantity of maple lumber for y modern rockers= 12*y board feet
Together this quantity should be less than or equal to 3000 board feet.
So, the final inequality is:
15*x+12*y <= 3000
Solving this equation to form a graph for the problem requires converting the equation to slope-intercept form as follows:
12*y <= 3000-15*xy <= 250-5/4*xthis can be rearranged to, y <=-5/4*x+250Here, 250 represents the y intercept on calculation we get 200 as the x intercept. Thus the area of the graph that this equation covers in the first quadrant can be represented by the these points:- (0, 250)----- Represents the y intercept- (200, 0)------Represents the x intercept- (0, 0)----------Represents the originThe region between these 3 points has to be the shaded region as that represents the set of solutions for the given inequality. The points outside this region (the triangular part between the three points) do not satisfy the equation and hence do not form the solutions to this inequality. The type of line used as the boundary has to be solid or continuous to represent the <= condition that this inequality stands for. The line will cut x axis on (0. 250) and y-axis on (200, 0) and will extend infinitely both ways.
Let us consider a point inside the shaded region: (1, 100) is the point that lies inside the shaded region,
Substituting the value in inequality we get,
15*1+12*100 <= 30001215 <= 300Since the above condition holds true, the point is a valid solution for the inequality and thus all points in the shaded region would satisfy the equation. It means 1 classic rocker and 12 modern rockers can be made in the given quantity of lumber and some would surely be left since the value for all points in the shaded region will always be less than the total quantity.
Let us consider a point on the line: (0, 250) is a point that lies on the line
15*0+12*250 <= 30003000 <= 3000Thus the condition is true. For all the values on the line, the condition would satisfy for the equal to part. In this example 0 classic rockers and 250 modern rockers would require exactly 3000 board feet of lumber and hence none will be left. This will be true for each point on the line.
Let us consider a point outside the shaded area: (250, 250)
15*250+12*250 <= 30006750 <= 3000Thus the in equation holds untrue for all values outside the shaded region. Here 250 modern rockers and 250 classic rockers require more than the available quantity of 3000 board feet of lumber and this is true for all values outside the shaded region. Question) 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? Solution) Quantity for lumber for classic rocker= 125*15= 1875 board feetQuantity of lumber for modern rockers= 175*12= 2100 board feetTotal Lumber required = 1875+ 2100= 3975 board feetTotal Quantity available = 3000 board feetTherefore, this shows that the order would need 3975-3000= 975 board feet of extra lumber and the total lumber that this order would require is 3975 board feet.