Two-Variable Inequality - Essay Example

Two variable Inequalities      Two variable Inequalities Introduction: Two variable inequalities de equations with two variables. The equations often take the form of ax + by = c, where a, b and c are real numbers and b and a represent values that cannot be zero simultaneously. This often presents a straight line and is thus a two variable linear inequality (Boyd, Laurent & Eric, 1994). Problem: 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:(1). Variable designations for different designs of rocker Ozark Furniture. This involves definition of variables for the different makes of chairs.a. Let (cmr) denote the number of classic maple rocking chairs andb. Let (mmr) denote the number of modern maple rocking chairs (2).

One classic maple rocking chair requires 15 board feet and this will be designated as (15 cmr) and since one modern maple rocking chair requires 12 board feet this requirement will be designated as follows (12 mmr). The totally available board feet of maple lumber is 3000 feet, therefore, an equation showing the total combination of classic maple rocking chairs and modern maple rocking chairs that could possibly be made out of this amount of materials is represented as shown in the following equation:15cmr + 12 mmr ≤ 3000(3).

Finding intercepts of the two variables on x axis (dependent variable) and y axis (independent variable) where either of them is zero, requires substitution of either of the variables with a zero value so as to obtain the maximum possible value of the other variable. The obtained intercepts can then provide end points on the line that will help draw a line that will map out the area of possibility. Cmr Intercept:Therefore, if cmr is on the horizontal axis and mmr is equals to = 0, then the equation becomes:15cmr ≤ 300015cmr / 15 ≤ 3000 / 15Cmr ≤ 200The cmr intercept is thus (200, 0) (cmr, mmr)Cmr Intercept:On the other hand, in order to find the mmr intercept, cmr’s value is equated to zero = 0 and the equation becomes: 12mmr ≤ 300012mmr / 12 ≤ 3000 / 12Mmr ≤ 250The mmr intercept is thus (0, 250) (cmr, mmr)The two end points on the intercepts represent the two ends of the line that maps out the region of possibility.

These points are (200, 0) and (0, 250)(4). Equation satisfaction region and inequality line. The inequality line in this case will be a straight, downward sloping solid line portraying a negative gradient, and the lines slopes from left to right because the inequality is a “less than or equal to” equation (Boyd et al., 1994). The region satisfying the equation falls within the first quadrant and is restricted to the left region of the line and the x and y axes (Boyd et al., 1994). (5). Arbitrary consideration of points within and without the region satisfying the equation.

Point within the region: (cmr, mmr) (75, 50)This point (75, 50) falls within the shaded region and it is a point at which the company can satisfy an order of 75 classic maple rocking chairs and 50 modern maple rocking chairs. If the company made 75 classic maple rocking chairs and 50 modern maple rocking chairs the amount of material used would have been:75 (15) + 50 (12) = 1725 The company would use 1725 feet of maple lumber and have 3000 – 1725 = 1275 feet of maple lumber as left over materialPoint without the region: (cmr, mmr) (150, 75)The point (150, 75) falls outside the equation satisfaction region and it is a point at which the company may be unable to produce the designated number of products required at this point.

If the company made 150 classic maple rocking chairs and 75 modern maple rocking chairs, the following amount of material would be required:150 (15) + 75 (12) = 3150The company would need to have 3150 feet of maple lumber to satisfy an order of 150 classic maple rocking chairs and 75 modern maple rocking chairs. This would present a (3150 – 3000 =150) 150 feet maple lumber shortage. The order cannot therefore, be fulfilled. Point on the line: (cmr, mmr) (100, 125)The point (100, 125) is exactly on the line and it is an implication that the company can fulfil the order using the exact amount of material available without any wastage or leftovers. 100 (15) + 125 (12) = 1500 + 1500 = 3000 At this point on the graph’s line the order is fulfilled, but there is no room for wastage or mistakes that could lead to wastage. (6). If a chain furniture store faxes an order for 175 modern rocking chairs and 125 classic rocking chairs, Ozark Furniture will be unable to satisfy the order because the order’s requirements fall outside the satisfaction region of the graph or equation.

This can be ascertained by performing a computation that would show the amount of required materials.(Cmr, mmr) (175, 125)175 (15) + 125 (12) = 2625 + 1500 = 4125Ozark would require 4125 feet of maple lumber to fulfil the order.4125 – 3000 = 1125 Therefore, there would be a shortage of 1125 feet of maple lumber, which would be necessary in order to fulfil the order. NOTE: In order to avoid using cmr and mmr because they appear lengthy, it would be appropriate to alternatively represent these two with just single letters such as c and m to get (c, m).

ReferencesBoyd, P. S. Laurent, E. and Eric, F. (1994),. Linear Matrix inequalities in System and Control Theory, Philadelphia: PA, SIAM (Society for Industrial and Applied Mathematics) Publishers