You have 60 megabytes (MB) of space left on your portable media player. You can choose to download song files that use 3.5 MB or video files that use 8 MB. You want to download at... You have 60 megabytes (MB) of space left on your portable media player. You can choose to download song files that use 3.5 MB or video files that use 8 MB. You want to download at least 12 files. What is a graph showing the numbers of song and video files you can download?
Understand the Problem
The question describes a scenario where you have a limited amount of storage space (60 MB) on a portable media player and need to download song files (3.5 MB each) and video files (8 MB each). You want to determine a graph that represents the possible combinations of song and video files you can download, given that you want to download at least 12 files in total.
Answer
The solution is a graph representing the following inequalities: $y \le -0.4375x + 7.5$ $y \ge -x + 12$ $x \ge 0$ $y \ge 0$
Answer for screen readers
The solution is a graph representing the following inequalities: $y \le -0.4375x + 7.5$ $y \ge -x + 12$ $x \ge 0$ $y \ge 0$
Steps to Solve
- Define variables
Let $x$ be the number of song files and $y$ be the number of video files.
- Set up the inequalities based on the given information
The total space used by the files must be less than or equal to 60 MB. Thus: $3.5x + 8y \le 60$.
The total number of files must be at least 12. Thus: $x + y \ge 12$.
Also, the number of files can't be negative $x \ge 0$ and $y \ge 0$.
- Solve for y in each inequality
For the space constraint: $3.5x + 8y \le 60$ $8y \le -3.5x + 60$ $y \le -\frac{3.5}{8}x + \frac{60}{8}$ $y \le -0.4375x + 7.5$
For the total number of files constraint: $x + y \ge 12$ $y \ge -x + 12$
- Determine the feasible region
The feasible region is defined by the intersection of the following inequalities: $y \le -0.4375x + 7.5$ $y \ge -x + 12$ $x \ge 0$ $y \ge 0$
The problem asks for the graph showing the numbers of songs and videos so you would graph the following inequalities: $y \le -0.4375x + 7.5$ $y \ge -x + 12$ $x \ge 0$ $y \ge 0$
The solution to the problem would be a graph of the region bounded by the above equations.
The solution is a graph representing the following inequalities: $y \le -0.4375x + 7.5$ $y \ge -x + 12$ $x \ge 0$ $y \ge 0$
More Information
The graph represents all possible combinations of song and video files that satisfy both the storage space constraint and the minimum number of files constraint. Any point $(x, y)$ within the feasible region of the graph represents a valid combination of $x$ song files and $y$ video files that can be downloaded.
Tips
A common mistake would be using = instead of $\le$ or $\ge$ Another common mistake would be to forget to include the constraints $x \ge 0$ and $y \ge 0$.
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