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

3 answers

To represent Anna's community service hours in a graphical format, we can define variables for each type of community service:

  • Let \( x \) represent the number of hours Anna spends on park beautification.
  • Let \( y \) represent the number of hours Anna spends on after-school child care.

From the problem statement, we know:

  • Each park beautification session earns her 4 hours, so \( x = 4p \) where \( p \) is the number of sessions with park beautification.
  • Each after-school child care session earns her 2 hours, so \( y = 2c \) where \( c \) is the number of sessions with after-school child care.

The total number of community service hours Anna needs to complete is at least 80 hours. Therefore, the inequality can be formulated as: \[ x + y \geq 80 \]

Substituting in the equations for \( x \) and \( y \): \[ 4p + 2c \geq 80 \]

To make it simpler for graphing purposes, let's divide the entire inequality by 2: \[ 2p + c \geq 40 \]

This equation indicates a linear relationship between the number of park beautification sessions (p) and the number of after-school child care sessions (c). The line \( 2p + c = 40 \) represents the boundary, and the area above this line (where \( 2p + c \geq 40 \)) represents all combinations of \( p \) and \( c \) where Anna meets or exceeds the 80-hour community service requirement.

In a graph:

  • The x-axis can represent the number of park beautification sessions (p).
  • The y-axis can represent the number of after-school child care sessions (c).
  • The line would slope down from the point (20, 0) on the p-axis (where 20 sessions of park beautification would give 80 hours) to (0, 40) on the y-axis (where 40 sessions of after-school child care would give 80 hours).

Look for a graph that contains a shaded area above the line \( 2p + c = 40 \) as that will indicate the values of \( p \) and \( c \) where Anna fulfills her community service requirement.

which one is the y and which one is the x axis

In the context of the problem, we can designate the axes as follows:

  • x-axis: This will represent the number of park beautification sessions (denote it as \(p\)).
  • y-axis: This will represent the number of after-school child care sessions (denote it as \(c\)).

So, in summary:

  • x-axis (horizontal): Number of park beautification sessions (p)
  • y-axis (vertical): Number of after-school child care sessions (c)

When plotting the line \(2p + c = 40\):

  • The intercept on the x-axis (when \(c = 0\)) is \(p = 20\), leading to the point (20, 0).
  • The intercept on the y-axis (when \(p = 0\)) is \(c = 40\), leading to the point (0, 40).

This configuration will allow you to visualize the combinations of sessions for both types of community service that meet or exceed the requirement of 80 hours. The area above the line \(2p + c = 40\) will represent the valid combinations for Anna to meet her community service hour requirement.