Question

Determine the 5 vertices of the feasible region, given the constraints below.
3x+5y≤30
x≤9
y≤3
x≥0
y≥0

Answer 1

The feasible region represents the set of points that satisfy all the given constraints.

1. To find the vertex where 3x + 5y = 30 intersects with x = 0, we set x = 0 and solve for y:
3(0) + 5y = 30
5y = 30
y = 6
So the first vertex is (0, 6).

2. To find the vertex where 3x + 5y = 30 intersects with y = 0, we set y = 0 and solve for x:
3x + 5(0) = 30
3x = 30
x = 10
However, x should be less than or equal to 9 according to the constraint x ≤ 9. So this point does not fall within the feasible region.

3. To find the vertex where x = 9 intersects with the line y = 0, we set x = 9 and y = 0:
3(9) + 5(0) = 27
So the second vertex is (9, 0).

4. To find the vertex where x = 9 intersects with the line y = 3, we set x = 9 and y = 3:
3(9) + 5(3) = 27 + 15 = 42
However, 42 is greater than 30 according to the constraint 3x + 5y ≤ 30. So this point does not fall within the feasible region.

5. To find the vertex where y = 3 intersects with the line x = 0, we set x = 0 and y = 3:
3(0) + 5(3) = 0 + 15 = 15
However, 15 is greater than 9 according to the constraint x ≤ 9. So this point does not fall within the feasible region.

Thus, the feasible region is bounded by the vertices (0, 6) and (9, 0).

Note: The constraints x ≥ 0 and y ≥ 0 indicate that the feasible region must be in the first quadrant, so the feasible region is the triangle formed by the points (0, 0), (0, 6), and (9, 0).