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.
Part B: Write an equation that represents the relationship. Show every step of your work.
Part C: Describe how you would graph the relationship. Use complete sentences.
Part D: How many points are awarded for 12 minutes of play?
To find the constant of proportionality, we can set up a proportion using the given information:
For every 4 minutes of play, 0.5 points are awarded.
For every 10 minutes of play, 1.25 points are awarded.
Let x be the number of minutes played and y be the number of points awarded.
We can set up the proportion as follows:
(0.5 points)/(4 minutes) = (1.25 points)/(10 minutes)
Now we can cross multiply:
0.5 * 10 = 1.25 * 4
5 = 5
Since both sides of the equation are equal, we have found the constant of proportionality, which is equal to 5.
For part B, we can use the equation:
y = 5x
This equation represents the relationship between the number of minutes played (x) and the number of points awarded (y).
For part C, we can graph the relationship by plotting points on a coordinate plane. The x-axis represents the number of minutes played and the y-axis represents the number of points awarded. We can choose different values for x (minutes played) and use the equation y = 5x to find the corresponding values of y (points awarded). Plotting these points and connecting them will give us a straight line.
For part D, we can use the equation y = 5x with x = 12 minutes to find the number of points awarded:
y = 5(12)
y = 60
Therefore, 60 points are awarded for 12 minutes of play.