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: To find the constant of proportionality, we can set up a proportion using the information given. We know that 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.

Let x represent the number of minutes played and y represent the number of points awarded.
We can set up the following proportion:

4 minutes / (1/2 point) = 10 minutes / (5/4 points)

To simplify the left side, we can multiply the numerator and denominator by 2:

8 minutes / 1 point = 10 minutes / (5/4 points)

To simplify the right side, we can multiply the numerator and denominator by 4/5:

8 minutes / 1 point = (10 minutes * 4/5) / 1 point

8 points = (40 minutes / 5) points

Now we can set up an equation to find the constant of proportionality:

8 = 40k / 5

Multiplying both sides of the equation by 5:

40 = 8k

Dividing both sides of the equation by 8:

5 = k

Therefore, the constant of proportionality is 5.

Part B: The equation that represents the relationship is y = 5x, where y represents the number of points awarded and x represents the number of minutes played.

Part C: To graph the relationship, we can create a coordinate plane with the x-axis representing the number of minutes played and the y-axis representing the number of points awarded. We can then plot points that represent different amounts of time played and the corresponding number of points awarded. By connecting these points with a straight line, we can visualize the linear relationship between the number of minutes played and the number of points awarded.

Part D: To find the number of points awarded for 12 minutes of play, we can substitute x = 12 into the equation y = 5x:

y = 5(12)
y = 60

Therefore, 60 points are awarded for 12 minutes of play.