Answer: Oki should make 0 bags of her recipe, and Stephen should make 30 bags of his recipe.
For vertex (15,15):
3x + 4y = 120 --> 3(15) + 4(15) = 120 --> x = 15
3x + 2y = 90 --> 3(15) + 2y = 90 -->
Step-by-step explanation: A. To find the maximum number of bags of trail mix that Oki and Stephen can make together, we can use a system of inequalities to represent the constraints:
Let "x" be the number of bags of trail mix that Oki makes and "y" be the number of bags that Stephen makes. Then we have:
3x + 4y ≤ 120 (constraint on the number of cups of nuts)
3x + 2y ≤ 90 (constraint on the number of cups of dried fruit)
x ≥ 0 (non-negative constraint for Oki's bags)
y ≥ 0 (non-negative constraint for Stephen's bags)
B. To graph the feasible region, we can start by graphing the two constraint equations as lines:
3x + 4y = 120 (line A)
3x + 2y = 90 (line B)
We can find the x and y intercepts for each line:
For line A:
When x = 0, 4y = 120, y = 30 (y-intercept)
When y = 0, 3x = 120, x = 40 (x-intercept)
For line B:
When x = 0, 2y = 90, y = 45 (y-intercept)
When y = 0, 3x = 90, x = 30 (x-intercept)
Next, we can shade the region that satisfies all of the constraints. This region is below line A and to the left of line B, and is bounded by the x and y axes.
We can find the vertices of the feasible region by finding the intersection points of the two lines and the axes. These vertices are (0,0), (0,30), (15,15), and (30,0).
C. To find the maximum number of bags of trail mix that Oki and Stephen can make together, we need to evaluate the objective function at each vertex of the feasible region, and choose the vertex that maximizes the objective function.
The objective function is the total number of bags of trail mix:
z = x + y
Evaluating at the vertices:
Vertex (0,0): z = 0 + 0 = 0
Vertex (0,30): z = 0 + 30 = 30
Vertex (15,15): z = 15 + 15 = 30
Vertex (30,0): z = 30 + 0 = 30
The maximum value of the objective function occurs at vertices (0,30) and (15,15), where z = 30 bags of trail mix.
To determine how many of each type of recipe to make, we can substitute each vertex into the two constraint equations to find the corresponding values of x and y.
For vertex (0,30):
3x + 4y = 120 --> 3x + 4(30) = 120 --> 3x = -90 --> x = -30
3x + 2y = 90 --> 3(-30) + 2(30) = 90 --> y = 15
Therefore, Oki should make 0 bags of her recipe, and Stephen should make 30 bags of his recipe.
For vertex (15,15):
3x + 4y = 120 --> 3(15) + 4(15) = 120 --> x = 15
3x + 2y = 90 --> 3(15) + 2y = 90 -->