Williams High School requires each student to complete at least 80 hours of community service before graduation.

Question

Anna, a student at Williams High School, decides to complete her community service hours with two different causes: park beautification and after-school child care. Each time Anna serves with a park beautification organization, she earns four hours of community service. Each time Anna serves with an after-school child care organization, she earns two hours of community service.

Which of the following graphs represents this situation?

J.
K.

L.
M.

Answer 1

To represent Anna's community service hours graphically, we need to create a graph where the x-axis represents the number of times she serves in park beautification (let's call this $x$) and the y-axis represents the number of times she serves in after-school child care (let's call this $y$).

Each service with park beautification earns her 4 hours, so the contribution of park beautification to her total hours can be represented by $4x$. Each service with after-school child care earns her 2 hours, so the contribution of after-school child care can be represented by $2y$.

The total hours of community service Anna needs to complete is 80 hours. Therefore, the equation we need to represent is:

$$4x+2y\ge 80$$

We can simplify this inequality to:

$$2x+y\ge 40$$

Interpreting the graph:

1. The line $2x+y=40$ will have intercepts that can be calculated:

   • If $x=0$ (no park beautification), then $y=40$ (20 after-school child care sessions).
   • If $y=0$ (no after-school child care), then $x=20$ (5 park beautification sessions).

2. The area above this line will represent the combinations of park beautification and after-school child care that satisfy Anna's requirement of at least 80 hours of community service.

Now, if you have multiple graphs (J, K, L, M) to choose from, look for a graph that has:

• A line with a downward slope intersecting the x-axis at 20 and y-axis at 40.
• A shaded area above the line, indicating the hours are greater than or equal to 80.

Once you identify a graph that meets this criteria, you will have found the correct representation of Anna's community service requirement.