a) Let ‘c’ be the number of classic maple rocking chairs and ‘m’ be the number of classic maple rocking chairs that Ozark Furniture Company make. b) Here we have given that a classic maple rocker requires 15 board feet of maple, and a modern rocker requires 12 board feet of maple. We have ‘c’ classic maple rocker, so total maple required for classic maple rocker chair = 15c board feet. And we have ‘m’ modern maple rocker, so total maple required for classic maple rocker chair = 12m board feet. We have total maple available is 3000 board feet.

Hence we can show this condition as:

15c + 12m ≤ 3000.

This is the inequality which is showing the given situation.

c) Graph for this inequality:

d) Here this is a less than and equal to inequality, so we need to take the area which is below to the line 15c + 12m = 3000 and this will be a solid line on the boundaries. The region lies between origin to x and y – intercepts. Each point which is in the first quadrant and under the shaded area is a solution for the given inequality. So, if (x, y) is a point in the shaded region then Ozark Furniture Company can obtain x number of classic and y number of modern maple rocking chairs. So, x – axis is showing number of classic maple rocking chairs and y – axis is showing the number of modern maple rocking chairs. i) Consider the point (50, 100).

This point is indicating that company is making 50 classic and 100 modern maple rocking chairs. So, c = 50 and m = 100.

So, 15c + 12m = 15(50) + 12(100) = 750 + 1200 = 1950.

Total board maple available = 3000 board feet.

Hence, total remaining maple = 3000 – 1950 = 1050 board feet. So, (50, 100) point, which is within the region showing that the company is making 50 classic and 100 modern maple rocking chairs and using 1950 board feet of maple. In this case total waste of maple is 1050 board feet. So, the company can fill the order easily. ii) Consider the point (200, 100).

This point is indicating that the company is making 200 classic and 100 modern maple rocking chairs. So, c = 200 and m = 100.

So, 15c + 12m = 15(200) + 12(100) = 3000 + 1200 = 4200.

Total board maple available = 3000 board feet.

Hence, extra maple required = 4200 – 3000 = 1200 board feet. So, (200, 100) point, which is out of the region showing that the company is making 200 classic and 100 modern maple rocking chairs and need 4200 board feet of maple. In this case the company needs 1200 board feet of maple. So, company cannot fill this order with the available maple. iii) Consider the point (100, 125).

This point is indicating that the company is making 100 classic and 125 modern maple rocking chairs. So, c = 100 and m = 125.

So, 15c + 12m = 15(100) + 12(125) = 1500 + 1500 = 3000.

Total board maple available = 3000 board feet.

So, here the company has exact amount of required maple.

So, (100, 125) point, which is exactly on the line showing that the company is making 100 classic and 125 modern maple rocking chairs and need to be very careful because company has exact amount of maple available to complete this order. So, company can complete the order. f) Here the order is of 125 classic rocking chairs and 175 modern rocking chairs. In terms of point we can show this condition as (125, 175). So, we have c = 125 and m = 175.

So, 15c + 12m = 15(125) + 12(175) = 1875 + 2100 = 3975 board feet. Total maple available = 3000 board feet.

So, Ozark Furniture will need 3975 – 3000 = 975 board feet of lumber. So, Ozark Furniture is not able to fill this order because the point is outside of the shaded area and company will need 975 board feet of lumber to fill the order. Conclusion: This was a real life example that how to use inequalities in our business. By the use of this we got to know that we will able to fill any order or not and if not then how much more raw-material we need to complete the order. This was very interesting question and we discussed all the 3 possible cases in this.

References

Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.

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