A small firm produces both AM and AM/FM car radios. The AM radios take 15 h
to produce, and the AM/FM radios take 20 h. The number of production hours is
limited to 300 h per week. The plant’s capacity is limited to a total of 18 radios per
week, and existing orders require that at least 4 AM radios and at least 3 AM/FM
radios be produced per week. Write a system of inequalities representing this situation.
Then, draw a graph of the feasible region given these conditions, in which x
is the number of AM radios and y the number of AM/FM radios. Con anybody help me with this problem? Please!!!!!!!
Of course! I can help you with this problem. Let's start by writing the system of inequalities to represent the given situation.
Let x be the number of AM radios produced per week, and y be the number of AM/FM radios produced per week.
1. The production time constraint: The AM radios take 15 hours to produce, and the AM/FM radios take 20 hours. The total production time is limited to 300 hours per week.
The inequality representing this constraint is: 15x + 20y ≤ 300.
2. The total number of radios constraint: The plant's capacity is limited to a total of 18 radios per week.
Inequality: x + y ≤ 18.
3. Existing order constraints: At least 4 AM radios and at least 3 AM/FM radios need to be produced per week.
Inequalities: x ≥ 4, y ≥ 3.
Now, let's graph the feasible region using these inequalities. To do this, we'll create a coordinate plane with x representing the number of AM radios (horizontal axis) and y representing the number of AM/FM radios (vertical axis).
Start by plotting the points where each inequality intersects the axes:
1. 15x + 20y ≤ 300:
- When x = 0, 20y ≤ 300 → y ≤ 15.
- When y = 0, 15x ≤ 300 → x ≤ 20.
Plot the points (0, 15) and (20, 0).
2. x + y ≤ 18:
- When x = 0, y ≤ 18.
- When y = 0, x ≤ 18.
Plot the points (0, 18) and (18, 0).
3. x ≥ 4 and y ≥ 3:
Plot the points (4, 0) and (0, 3).
Now, connect these points to form the feasible region, which will be the area that satisfies all the inequalities. Shade this region on the graph.
I hope this explanation helps you understand how to solve this problem. If you have any further questions, feel free to ask!