Find the minimum value of
C = 3x + 10y
subject to the following constraints:
{2x+4y≥20
{2x + 2y≤16
{x>2
{y≥3
To find the minimum value of C, we need to find the values of x and y that satisfy all the constraints and minimize the expression 3x + 10y.
1. Constraint: 2x + 4y ≥ 20
Rearranging this constraint, we get:
2x ≥ 20 - 4y
x ≥ 10 - 2y
2. Constraint: 2x + 2y ≤ 16
Rearranging this constraint, we get:
2y ≤ 16 - 2x
y ≤ 8 - x
3. Constraint: x > 2
This constraint states that x must be greater than 2, so it does not affect the possible values of x and y.
4. Constraint: y ≥ 3
This constraint states that y must be greater than or equal to 3, so it does not affect the possible values of x and y.
To find the minimum value of C, we need to find the intersection of these constraints.
Plotting the inequalities on a graph, we find that the feasible region is a triangular region bounded by the lines x = 10 - 2y, y = 8 - x, y = 3, and x = 2.
We need to find the point within this region that minimizes the expression C = 3x + 10y.
Using substitution, we can write C in terms of a single variable, either x or y.
From equation 1: x ≥ 10 - 2y, we can solve for x and substitute it into C:
x = 10 - 2y
C = 3(10 - 2y) + 10y
C = 30 - 6y + 10y
C = -6y + 30
To minimize C, we need to find the lowest possible value of y within the feasible region.
The lowest value of y is at the point where the line y = 3 intersects the feasible region.
Substituting y = 3 into equation 1:
x ≥ 10 - 2(3)
x ≥ 10 - 6
x ≥ 4
Thus, the minimum value of C occurs when x = 4 and y = 3.
Plugging these values into the expression for C:
C = -6(3) + 30
C = -18 + 30
C = 12
Therefore, the minimum value of C is 12.