Question

A manufacturer produces two models of toy airplanes. It takes the manufacturer 4 minutes to assemble model A and 8 minutes to package it. it takes the. manufacturer 7 minutes to assemble model B and 3 minutes to package it. In a given week, the total avalable time for assembling is 560 minutes, and the total avallable time for packaging is 480 minutes. Let x be the number of modet A units produced. Let y be the number of model B units produced. Shade the region corresponding to all values of x and y that satisfy these requirements.

Answer 1

To shade the region corresponding to all values of x and y that satisfy the given requirements, we need to create a system of inequalities based on the available assembly and packaging times and the time required for each model.

Let's define the following variables:

x: the number of Model A units produced.

y: the number of Model B units produced.

The time spent on assembling Model A and Model B is as follows:

Model A assembly time: 4 minutes per unit.

Model B assembly time: 7 minutes per unit.

The time spent on packaging Model A and Model B is as follows:

Model A packaging time: 8 minutes per unit.

Model B packaging time: 3 minutes per unit.

We have the following constraints based on the available time for assembling and packaging:

The total available assembly time is 560 minutes: 4x (Model A) + 7y (Model B) ≤ 560.

The total available packaging time is 480 minutes: 8x (Model A) + 3y (Model B) ≤ 480.

These are the inequalities that define the shaded region. To find this region, you would typically graph the lines corresponding to these inequalities and shade the area where they overlap. However, I'm unable to draw graphs, so I can't provide a visual representation. You can use graph paper or graphing software to plot these lines and find the shaded region where both conditions are met. This region represents all valid combinations of x and y values that satisfy the given time constraints.

Answer 2

Explanation:

To visualize the region corresponding to the values of x and y that satisfy the given requirements, we can create a graph. The x-axis will represent the number of model A units produced (x), and the y-axis will represent the number of model B units produced (y).

First, let's calculate the time required for assembling and packaging for each model:

Model A:

Time for assembling = 4 minutes/unit

Time for packaging = 8 minutes/unit

Model B:

Time for assembling = 7 minutes/unit

Time for packaging = 3 minutes/unit

Now, let's consider the constraints given in the problem:

1. Total available time for assembling: 560 minutes

This means that the total time spent on assembling both model A and model B should not exceed 560 minutes. We can express this constraint as:

4x + 7y ≤ 560

2. Total available time for packaging: 480 minutes

This means that the total time spent on packaging both model A and model B should not exceed 480 minutes. We can express this constraint as:

8x + 3y ≤ 480

To shade the region that satisfies these requirements, we need to find the intersection of the lines represented by the above inequalities.

First, let's plot the lines:

For the inequality 4x + 7y ≤ 560:

- Set x = 0: 7y ≤ 560 → y ≤ 80

- Set y = 0: 4x ≤ 560 → x ≤ 140

For the inequality 8x + 3y ≤ 480:

- Set x = 0: 3y ≤ 480 → y ≤ 160

- Set y = 0: 8x ≤ 480 → x ≤ 60

Now, let's plot the lines x = 0, y = 0, x = 140, and y = 80 on the graph. The shaded region will be below the line x = 140, to the left of the line y = 80, and within the intersection of the shaded regions of the two inequalities.

The graph will show the feasible region where the values of x and y satisfy the given requirements.