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.

Question

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. (4 points)

Part B: Write an equation that represents the relationship. Show every step of your work. (2 points)

Part C: Describe how you would graph the relationship. Use complete sentences. (4 points)

Part D: How many points are awarded for 12 minutes of play? (2 points)

Answer 1

Part A:

We can set up a proportion to find the constant of proportionality. Let's call the constant of proportionality "k".

For every 4 minutes of play, the game awards one-half point.
So, we have the proportion: 4 minutes / x points = 1/2 point / 1 game minute

Cross multiplying, we get: 4 minutes * 1 game minute = x points * 1/2 point

Simplifying, we have: 4 = x/2

To isolate x, we can multiply both sides of the equation by 2.

2 * 4 = 2 * (x/2)

8 = x

Therefore, the constant of proportionality is 8.

Part B:

Now that we know the constant of proportionality is 8, we can write the equation that represents the relationship.

Let's call the time played "t" in minutes and the points awarded "p".

The equation would be: p = 8t

Part C:

To graph the relationship, we would plot points on a graph using the time played, t, as the x-axis and the points awarded, p, as the y-axis.

We can choose different values for t, plug them into the equation, and calculate the corresponding values of p. By doing this for multiple values, we will be able to plot points on the graph.

For example, if we choose t = 4, we can calculate p as follows:
p = 8 * 4 = 32

So, for 4 minutes of play, the game awards 32 points.

We can continue this process for other values of t, such as 8, 12, 16, etc., to plot more points on the graph. By connecting these points, we will get a straight line that represents the relationship between time played and points awarded.

Part D:

To find the number of points awarded for 12 minutes of play, we can plug 12 into the equation we derived in Part B.

p = 8t
p = 8 * 12
p = 96

Therefore, 12 minutes of play would earn the player 96 points.