Rocky Balboa, the boxing legend, and his rival Ivan Drago decided to have a friendly competition to see who could throw the most punches in a given time period. They agreed to throw a combination of jabs and hooks, with jabs counting as 1 point and hooks counting as 2 points. Rocky can throw an average of 30 punches per minute, while Ivan can throw 25 punches per minute. They each have 3 minutes to throw punches. Rocky wants to throw the most punches and earn the most points in the competition. If x represents the number of jabs Rocky throws and y represent the number of hooks he throws, write and solve a system of inequalities graphically and determine one possible solution.

Answer 1

Let's first determine the equations based on the given information:

1. Rocky can throw an average of 30 punches per minute: x + 2y = 30
2. Ivan can throw an average of 25 punches per minute: x + 2y ≤ 25

Since Rocky has 3 minutes to throw punches, the total number of punches he can throw must be less than or equal to 90.

3x + 3(2y) ≤ 90
3x + 6y ≤ 90
x + 2y ≤ 30

Now, we have the following system of inequalities:
x + 2y = 30
x + 2y ≤ 25
x + 2y ≤ 30

Let's graph these inequalities:

graph{x+2y=30 [-10, 40, -10, 40]}

graph{x+2y<=25 [-10, 40, -10, 40]}

graph{x+2y<=30 [-10, 40, -10, 40]}

From the graph, the feasible region where all three inequalities overlap is the shaded region below the line x + 2y = 30 and to the left of the line x + 2y ≤ 25.

One possible solution within this region could be x = 10 and y = 10. This means Rocky throws 10 jabs and 10 hooks, earning 30 points in total.

Therefore, one possible solution is Rocky throwing 10 jabs and 10 hooks within the given time period to earn the most points in the competition.