Find the value(s) of the function, subject to the system of inequalities.
Find the maximum and minimum of P = 10x - 16y subject to: 0 ¡Ü x ¡Ü 5, 0 ¡Ü y ¡Ü 8, 4x + 5y ¡Ü 30, and 4x + 3y ¡Ü 20
50, 0
-96, 0
-67.5, -96
50, -96
To solve this problem, we first need to find the feasible region represented by the system of inequalities. The feasible region is the region that satisfies all the inequalities.
1. Start by graphing the lines determined by the inequalities:
- 0 ≤ x ≤ 5 defines a vertical line segment from (0,0) to (5,0).
- 0 ≤ y ≤ 8 defines a horizontal line segment from (0,0) to (0,8).
- 4x + 5y ≤ 30 defines a line with a slope of -4/5 passing through (0,6) and (7.5,0)
- 4x + 3y ≤ 20 defines a line with a slope of -4/3 passing through (0,6.67) and (5,0)
2. Shade the region that satisfies all the inequalities. The region below both lines represents the feasible region.
3. Next, evaluate the function P = 10x - 16y at the vertices of the shaded feasible region:
- (0,0): P = 10(0) - 16(0) = 0
- (5,0): P = 10(5) - 16(0) = 50
- (0,8): P = 10(0) - 16(8) = -128
- (7.5,0): P = 10(7.5) - 16(0) = 75
- (5, ((20/3) - (16/3))): P = 10(5) - 16((20/3) - (16/3))) = -32
- (((20/4) - (5/4)), 8): P = 10(((20/4) - (5/4))) - 16(8) = -62
4. Now we can determine the maximum and minimum values of P:
- The maximum value of P is 75, which occurs at the vertex (7.5,0).
- The minimum value of P is -128, which occurs at the vertex (0,8).
Therefore, the correct answer is: (-96, 0)
To find the maximum and minimum values of the function P = 10x - 16y subject to the system of inequalities, we need to find the feasible region determined by the intersection of the inequalities and then evaluate the function at each extreme point of the feasible region.
First, let's plot the system of inequalities on a graph to visualize the feasible region.
The inequalities are:
0 ≤ x ≤ 5
0 ≤ y ≤ 8
4x + 5y ≤ 30
4x + 3y ≤ 20
First, let's plot the lines for the two equations:
4x + 5y = 30
4x + 3y = 20
To graph them, we can rewrite the equations in slope-intercept form (y = mx + b):
4x + 5y = 30 -> y = (-4/5)x + 6
4x + 3y = 20 -> y = (-4/3)x + 20/3
Now, let's graph these lines on a coordinate plane:
First, plot the line 4x + 5y = 30 by plotting two points on the line. For example, when x = 0, y = 6, and when y = 0, x = 7. Connect these two points with a line.
Next, plot the line 4x + 3y = 20 by plotting two points on the line. For example, when x = 0, y = 20/3, and when y = 0, x = 5. Connect these two points with a line.
Now, let's shade the feasible region that satisfies all the inequalities. The region is bounded by the lines and the given constraints: 0 ≤ x ≤ 5 and 0 ≤ y ≤ 8.
The shaded region represents the feasible region.
We want to find the maximum and minimum values of the function P = 10x - 16y within this feasible region.
To find the extreme points of the feasible region, we can evaluate the function P at the vertices of the region. These vertices are defined by the intersection points of the lines and the constraints.
By evaluating P = 10x - 16y at each vertex, we can determine the maximum and minimum values.
Based on the graph, one of the vertices of the feasible region is at (0, 0). Evaluating P = 10x - 16y at this point gives:
P = 10(0) - 16(0) = 0
Another vertex is at (5, 0). Evaluating P at this point gives:
P = 10(5) - 16(0) = 50
The third vertex is at (5, 6). Evaluating P at this point gives:
P = 10(5) - 16(6) = -67.5
The fourth vertex is at (0, 8). Evaluating P at this point gives:
P = 10(0) - 16(8) = -96
Therefore, the maximum and minimum values of P are 50 and -96, respectively. Hence, the correct answer is option D: 50, -96.