Teddy Bear Problem Paper

In this problem, we are starting our own toy-making business. However, we are only able to produce two different toys out of five options (dolls, rocking horses, tinker toys, hammocks, and teddy bears). We have materials and time on hand (1000 feet of wood, 1000 yards of string, 400 pints of paint, and 600 yards of fabric in 720 hours) and we cannot afford to buy more. We make $25.50 from dolls, $37.40 from rocking horses, $19.55 from tinker toys, $43.35 from hammocks, and $22.10 from teddy bears. Our task is to figure out which two toys to choose and the amount of the toys we need to make for maximum profit. We decided to begin this problem by setting up a linear programming problem for each set of two toys. There are 10 sets of toys that

First, we set up the equation for the profit. Since rocking horses made $37.40 and teddy bears made $22.10, the equation was 37.40x + 22.10y = P (x being rocking horses, y being teddy bears, and P being profit). Rocking horses require 10 feet of wood, 1 yard of string, 4 pints of paint, and they take 7 hours to make. Teddy bears require 2 yards of string, 1 pint of paint, 5 yards of fabric, and they take 5 hours to make. We then set up inequalities showing that we had to keep the amount of wood under 1,000 feet (10x ≤ 1,000), string under 1,000 yards (x + 2y ≤ 1,000), paint under 400 pints (4x + y ≤ 400), fabric under 600 yards (5y ≤ 600), and hours under 720 (7x + 5y ≤ 720). We converted all of those equations into slope-intercept form and graphed each of them. The end result was a profit of $3289.80. (See Figure 1 in the Appendix for

Our profit equation was 37.4y+43.35x=P. y represents the rocking horses, x represents the hammocks, P represents the profit, $37.40 is how much one rocking horse costs, and $43.35 is how much one hammock costs. The rocking horses require 10 feet of wood, 1 yard of string, 4 pints of paint, and 7 hours of preparation to make. The hammocks require 1 foot of wood, 20 yards of string, 0.5 pints of paint, and 4 hours of preparation to make. There are only 1,000 feet of wood, 1,000 yards of string, 400 pints of paint, 600 yards of fabric, and 720 hours available on hand. Next, we made a list of inequalities that represent the constraints of the problem (10y+x≤1,000; y+20x≤1,000; 4y+0.5x≤400; 0≤600; 7y+4x≤720). We put these inequalities into slope-intercept form so we are able to graph them. After graphing the inequalities, we shaded in the area of the graph where all answers could be true. To find this region, we plugged in a coordinate into all of the inequalities. This could be any coordinate, but we decided to choose (0,0). The statements were all true, so we shaded in the side of the graph where that coordinate was. After finding the shaded region, we identified the corner points of the shaded region. Next, we plugged in each of those points or coordinates into the profit equation. The point that gave us the highest profit was (45,77) and gave us a profit of $4,830.55. (See figure 3