Question

A factory can produce two products, x and y, with a profit approximated by P = 14x + 22y – 900. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1600. What production levels yield maximum profit? (1 point) Responses x = 400 y = 600 x = 400 y = 600 x = 0 y = 0 x = 0 y = 0 x = 1600 y = 0 x = 1600 y = 0 x = 0 y = 200

Answer 1

To find the production levels that yield the maximum profit, we need to solve the given system of inequalities.

x + 2y ≤ 1600 (restricting the production levels)
y - x ≤ 200 (limiting the difference between y and x)

We can solve this system by graphing or using linear programming techniques. Let's use graphing in this case.

First, let's graph the first inequality x + 2y ≤ 1600:

Plotting the x and y-intercepts:
To find the x-intercept, set y = 0:
x + 2(0) = 1600
x = 1600
So, the x-intercept is (1600, 0).

To find the y-intercept, set x = 0:
0 + 2y = 1600
2y = 1600
y = 800
So, the y-intercept is (0, 800).

Plotting these two points, we can draw a line that passes through them.

Next, let's graph the second inequality y - x ≤ 200:

Rewrite the inequality as y ≤ x + 200.

Plotting the x and y-intercepts:
To find the x-intercept, set y = 0:
0 ≤ x + 200
x ≥ -200
So, the x-intercept is (-200, 0).

To find the y-intercept, set x = 0:
y ≤ 0 + 200
y ≤ 200
So, the y-intercept is (0, 200).

Plotting these two points, we can draw a line that passes through them.

Now, we shade the region that satisfies both inequalities, which is the overlapping region between the two lines.

In this shaded region, we need to find the production levels that maximize the profit function P = 14x + 22y - 900.

To do that, we can evaluate the profit function at the vertices of the shaded region.

The vertices are (0, 0), (1600, 0), and (400, 600).

Evaluating the profit function at these vertices:

P(0, 0) = 14(0) + 22(0) - 900 = -900
P(1600, 0) = 14(1600) + 22(0) - 900 = 22,400 - 900 = 21,500
P(400, 600) = 14(400) + 22(600) - 900 = 5,600 + 13,200 - 900 = 18,900

The maximum profit is achieved at (400, 600) with a profit of 18,900.

So, the production levels that yield maximum profit are x = 400 and y = 600.