The minimum value of C is 72 when x = 2 and y = 2.
To find the minimum value of C = 10x + 26y, subject to the given constraints, we need to use a method called linear programming. This method involves finding the feasible region and then evaluating the objective function at the corner points of that region.
Let's graph the constraints on a coordinate plane to visualize the feasible region:
1. x + y ≤ 6: This is a straight line with a slope of -1 passing through (0, 6) and (6, 0). Shade the region below this line.
2. 5x + y ≥ 10: This is a straight line with a slope of -5 passing through (2, 0) and (0, 10). Shade the region above this line.
3. x + 5y ≥ 14: This is a straight line with a slope of -1/5 passing through (14, 0) and (0, 2.8). Shade the region above this line.
The feasible region is the overlapping shaded region.
Next, we need to find the corner points of this region. These are the points where the lines intersect:
1. Intersection of the lines x + y = 6 and 5x + y = 10: (1, 5)
2. Intersection of the lines x + y = 6 and x + 5y = 14: (4, 2)
3. Intersection of the lines 5x + y = 10 and x + 5y = 14: (2, 2)
Now, substitute these points into the objective function C = 10x + 26y:
1. C = 10(1) + 26(5) = 10 + 130 = 140
2. C = 10(4) + 26(2) = 40 + 52 = 92
3. C = 10(2) + 26(2) = 20 + 52 = 72
So, the minimum value of C is 72 when x = 2 and y = 2.