Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
The first quadrant of a coordinate plane is shown. The x axis goes from 0 to 10 in intervals of 1 and the y axis goes from 0 to 10 in intervals of 1. Three points are plotted at left parenthesis 0 comma 8 right parenthesis, left parenthesis 5 comma 4 right parenthesis, and left parenthesis 9 comma 0 right parenthesis. The three points are connected with lines. The area under these lines, above the x axis, and to the right of the y axis is shaded.
Find the values of x and y that maximize the objective function P=3x+2y
for the graph. What is the maximum value?
To find the values of x and y that maximize the objective function P = 3x + 2y, we need to determine the coordinates of the vertex of the shaded region.
First, let's plot the three given points on the coordinate plane:
- Point A: (0, 8)
- Point B: (5, 4)
- Point C: (9, 0)
Next, let's draw lines connecting these three points.
Now, observe the shaded region under these lines, above the x-axis, and to the right of the y-axis.
To find the vertex of this region, we need to find the intersection point of two lines that form the border of the shaded region.
From point A (0, 8) and point C (9, 0), we can determine the equation of the line that passes through these two points.
The slope (m1) of this line is given by:
m1 = (y2 - y1) / (x2 - x1)
= (0 - 8) / (9 - 0)
= -8/9
Using the point-slope form of a line y - y1 = m(x - x1) with A(0, 8) as (x1, y1):
y - 8 = (-8/9)(x - 0)
y - 8 = (-8/9)x
y = (-8/9)x + 8
Similarly, from point B (5, 4) and point C (9, 0), we can determine the equation of the line that passes through these two points.
The slope (m2) of this line is given by:
m2 = (y2 - y1) / (x2 - x1)
= (0 - 4) / (9 - 5)
= -4/4
= -1
Using the point-slope form of a line y - y1 = m(x - x1) with B(5, 4) as (x1, y1):
y - 4 = (-1)(x - 5)
y - 4 = -x + 5
y = -x + 9
Next, we need to find the intersection point of these two lines by setting them equal to each other and solving for x and y:
(-8/9)x + 8 = -x + 9
Multiplying through by 9 to eliminate fractions:
-8x + 72 = -9x + 81
Rearranging the equation:
9x - 8x = 81 - 72
x = 9
Substituting x = 9 into one of the equations to solve for y:
y = -x + 9
y = -(9) + 9
y = 0
Therefore, the vertex of the shaded region (and the maximum point for P) is (9, 0).
To find the maximum value of P, substitute the values of x and y into the objective function P = 3x + 2y:
P = 3(9) + 2(0)
P = 27
Thus, the maximum value of P is 27.
To find the values of x and y that maximize the objective function P = 3x + 2y, we need to determine the vertex of the shaded region.
From the given information, we have three points plotted: (0, 8), (5, 4), and (9, 0). These three points represent the vertices of a triangle.
Step 1: Plot the coordinates on the graph.
- Plot the points (0, 8), (5, 4), and (9, 0) on the coordinate plane.
- Connect these points with lines to form a triangle.
Step 2: Find the intersection points.
- The intersection points of the lines are where the triangle vertices meet.
Step 3: Determine the vertex of the shaded region.
- The vertex of the shaded region will be the point where the objective function is maximized.
- This point will lie on one of the intersection points of the lines.
Step 4: Calculate the objective function at the vertex.
- Substitute the x and y values from the vertex into the objective function P = 3x + 2y to find the maximum value.
Let's calculate the steps:
Step 1:
- Plot the points (0, 8), (5, 4), and (9, 0) on the coordinate plane.
- Connect these points with lines to form a triangle.
Step 2:
- Find the intersection points. In this case, the lines intersect at (3, 6) and (7, 2).
Step 3:
- Determine the vertex of the shaded region. The vertex will be the point on one of the intersection points.
Step 4:
- Calculate the objective function at the vertex. Substitute the x and y values from the vertex into the objective function P = 3x + 2y.
- For the point (3, 6): P = 3(3) + 2(6) = 9 + 12 = 21.
- For the point (7, 2): P = 3(7) + 2(2) = 21 + 4 = 25.
Therefore, the values of x and y that maximize the objective function are x = 7 and y = 2, and the maximum value of P is 25.
To find the values of x and y that maximize the objective function P = 3x + 2y, we need to locate the point on the graph where the shaded region reaches its maximum value.
First, let's analyze the coordinates of the three points plotted on the graph: (0, 8), (5, 4), and (9, 0).
Now, we need to find the equation of the line connecting these points. Let's start with the line connecting (0, 8) and (5, 4):
Slope (m) = (y2 - y1) / (x2 - x1)
= (4 - 8) / (5 - 0)
= -4 / 5
= -0.8
Using the point-slope form, we have:
y - y1 = m(x - x1)
y - 8 = -0.8(x - 0)
y - 8 = -0.8x
Simplifying the equation, we get:
y = -0.8x + 8
Now, let's find the equation of the line connecting (5, 4) and (9, 0):
Slope (m) = (y2 - y1) / (x2 - x1)
= (0 - 4) / (9 - 5)
= -4 / 4
= -1
Using the point-slope form again, we have:
y - y1 = m(x - x1)
y - 4 = -1(x - 5)
y - 4 = -x + 5
y = -x + 9
We now have two equations:
y = -0.8x + 8
y = -x + 9
To find the maximum value of P = 3x + 2y within the shaded region, we should examine the intersection points of these two lines and calculate the corresponding P values.
Let's solve the system of equations by setting the y values equal to each other:
-0.8x + 8 = -x + 9
Simplifying the equation, we get:
0.2x = 1
x = 1 / 0.2
x = 5
Substituting this value of x into either of the equations, we find:
y = -0.8(5) + 8
y = -4 + 8
y = 4
Therefore, the point (x, y) that maximizes the objective function P = 3x + 2y is (5, 4).
Now, let's calculate the maximum value of P:
P = 3(5) + 2(4)
P = 15 + 8
P = 23
Hence, the maximum value of P is 23.