Mrs. Brown has a reward program in her classroom for students who have perfect or good weekly conduct. Each student that earns perfect conduct for a week gets three gold stars, and each student that earns good conduct for a week gets two gold stars. At the end of each six-weeks, Mrs. Brown provides breakfast for all students that have earned more than 12 gold stars.

Question

Which of the following graphs represents the number of perfect conduct weeks and good conduct weeks students need?

GPT-4o mini · Answer

To determine how many weeks of perfect conduct (which earns 3 gold stars each week) and good conduct (which earns 2 gold stars each week) are needed to surpass the 12 gold stars required for breakfast, we can set up an equation.

Let:

$ p $ = number of weeks of perfect conduct
$ g $ = number of weeks of good conduct

The total number of gold stars can be represented by the equation: \[ 3p + 2g > 12 \]

To visualize this, we can rearrange the equation to find the boundary where students start to earn more than 12 gold stars: \[ 3p + 2g = 12 \]

Now, let’s find some points to graph this equation:

If $ p = 0 $: $ 2g = 12 $ → $ g = 6 $ (Point: (0, 6))
If $ p = 4 $: $ 3(4) + 2g = 12 $ → $ 12 + 2g = 12 $ → $ g = 0 $ (Point: (4, 0))
If $ g = 0 $: $ 3p = 12 $ → $ p = 4 $ (Point: (4, 0))
If $ g = 3 $: $ 3p + 2(3) = 12 $ → $ 3p + 6 = 12 $ → $ 3p = 6 $ → $ p = 2 $ (Point: (2, 3))

Plotting these points and the line they form:

The line will intercept the x-axis at (4, 0) and the y-axis at (0, 6).

To determine the area representing the students who qualify for breakfast, shade the area above the line $ 3p + 2g = 12 $ because we are looking for cases where the total gold stars exceed 12.

In summary, you want to look for a graph that shows a line starting from (0, 6) to (4, 0) and then shade the area above this line to represent students earning more than 12 gold stars. If you should choose between different options, select the one that accurately represents this scenario and shading.

Answer

is the line dashed or not

GPT-4o mini · Answer

The line representing the equation $3p + 2g = 12$ should be dashed. This is because the condition is strict: students need to earn **more than** 12 gold stars to qualify for breakfast (i.e., $3p + 2g > 12$). A dashed line indicates that the points on the line itself (where $3p + 2g = 12$) are not included in the solution set. Therefore, the area above the dashed line represents the combinations of perfect and good conduct weeks where students earn more than 12 gold stars.