PLEASE HELP!! SHOW WORK
A factory can produce two products, x and y, with a profit approximated by P = 14x + 22y – 900. The production of y must exceed the production of x by at least 100 units. Moreover, production levels are limited by the formula x + 2y ≤ 1400.
Identify the vertices of the feasible region.
What production levels yield the maximum profit, and what is the maximum profit?
To find the vertices of the feasible region, we need to graph the inequalities and find the points where the lines intersect. Let's solve the equations step by step to find the vertices:
1. Start with the inequality x + 2y ≤ 1400:
Rewrite the inequality in slope-intercept form:
2y ≤ -x + 1400
Divide both sides by 2:
y ≤ -0.5x + 700
Graph this line on a coordinate system:
Plot the y-intercept at (0, 700) and use the slope to find additional points.
For example, when x = 2000, y = -0.5(2000) + 700 = -600 + 700 = 100.
Join the points to form a boundary line.
2. Next, consider the condition that production of y must exceed the production of x by at least 100 units:
y ≥ x + 100
Graph this line using a similar process as above.
Plot the y-intercept at (0, 100) and use the slope to find additional points.
3. Find the points where the two boundary lines intersect. These are the vertices of the feasible region.
Solve the system of equations by setting the two boundary lines equal to each other:
-0.5x + 700 = x + 100
1.5x = 600
x = 400
Substitute this value back into either equation to find y:
y = 400 + 100 = 500
The first vertex is (400, 500).
4. Repeat the process to find the other vertices.
We can use the same method to find the next intersection between lines by solving the system:
-0.5x + 700 = x + 100
1.5x = 600
x = 400
y = 400 + 100 = 500
The second vertex is (400, 500).
The third vertex can be found where the x-axis intersects with the line y = -0.5x + 700:
-0.5x + 700 = 0
-0.5x = -700
x = 1400
The third vertex is (1400, 0).
Now we have the vertices of the feasible region: (400, 500), (400, 500), and (1400, 0).
To find the production levels that yield the maximum profit, we substitute the x and y values of each vertex into the profit function P = 14x + 22y - 900 and see which one gives the highest result.
For the first vertex: P = 14(400) + 22(500) - 900 = 5600 + 11000 - 900 = 16400
For the second vertex: P = 14(400) + 22(500) - 900 = 5600 + 11000 - 900 = 16400
For the third vertex: P = 14(1400) + 22(0) - 900 = 19600 - 900 = 18700
The maximum profit is $18,700, and it is achieved by producing 1400 units of x and 0 units of y.
In google paste:
A factory can produce two products, x and y, with a profit approximated by P = 14x + 22y – 900. The production of y must exceed the production of x by at least 100 units. Moreover, production levels are limited by the formula x + 2y ≤ 1400.
When you see list of results go on:
h t t p s:/ /b ainly.c o m › High School › Mathematics
You will see answer with explanation.