Question
Choose a point that falls in the shaded region for both this inequality and the inequality you found in Task 1. Explain what the x-coordinate and y-coordinate represent and the significance in terms of both cost and money earned of this point falling in the shaded regions.
Answers
GPT 3.5
Let's consider the point (3, 5), which falls in the shaded region for both inequalities:
1. Inequality 1: 3x + 4y <= 24
When we substitute x = 3 and y = 5 into this inequality, we have:
3(3) + 4(5) <= 24
9 + 20 <= 24
29 <= 24 (which is false)
Therefore, it does not satisfy the first inequality.
2. Inequality 2: 2x + 3y >= 12
When we substitute x = 3 and y = 5 into this inequality, we have:
2(3) + 3(5) >= 12
6 + 15 >= 12
21 >= 12 (which is true)
Therefore, it satisfies the second inequality.
Explanation:
- In this context, the x-coordinate represents the number of hours spent on research, while the y-coordinate represents the number of hours spent tutoring.
- The significance of the point (3, 5) falling in the shaded regions of both inequalities shows that if one spends 3 hours on research and 5 hours tutoring, they are meeting the minimum requirement to earn at least $12, while not exceeding the maximum cost of $24.
- This point represents a balance between cost and money earned, as it ensures that the tutoring sessions are profitable while maintaining research activities within budget constraints.
1. Inequality 1: 3x + 4y <= 24
When we substitute x = 3 and y = 5 into this inequality, we have:
3(3) + 4(5) <= 24
9 + 20 <= 24
29 <= 24 (which is false)
Therefore, it does not satisfy the first inequality.
2. Inequality 2: 2x + 3y >= 12
When we substitute x = 3 and y = 5 into this inequality, we have:
2(3) + 3(5) >= 12
6 + 15 >= 12
21 >= 12 (which is true)
Therefore, it satisfies the second inequality.
Explanation:
- In this context, the x-coordinate represents the number of hours spent on research, while the y-coordinate represents the number of hours spent tutoring.
- The significance of the point (3, 5) falling in the shaded regions of both inequalities shows that if one spends 3 hours on research and 5 hours tutoring, they are meeting the minimum requirement to earn at least $12, while not exceeding the maximum cost of $24.
- This point represents a balance between cost and money earned, as it ensures that the tutoring sessions are profitable while maintaining research activities within budget constraints.