For a certain video game, the number of points awarded to the player is proportional to the amount of time the game is played. For every 4 minutes of play, the game awards one-half point, and for every 10 minutes of play, the game awards one and one-fourth points.
Part A: Find the constant of proportionality. Show every step of your work.
Let's assume the time played is represented by "t" and the points awarded is represented by "p". We want to find the constant of proportionality, which we'll call "k".
According to the problem, for every 4 minutes of play, the game awards one-half point. This can be represented as:
p = (1/2)(t/4)
Simplifying this equation, we get:
p = (1/2)(1/4)t
p = (1/8)t
Now, let's consider the second condition given in the problem. For every 10 minutes of play, the game awards one and one-fourth points. This can be represented as:
p = (5/4)(t/10)
Simplifying this equation, we get:
p = (5/4)(1/10)t
p = (1/8)t
We can see that the two equations we derived are the same, which means the constant of proportionality is equal to 1/8.
Therefore, the constant of proportionality is 1/8.