Answer :
Let's go through each part of the question step-by-step.
### Part A: Find the constant of proportionality
1. From the problem, we know:
- For every 4 minutes of play, the game awards [tex]\(\frac{1}{2}\)[/tex] point.
- For every 12 minutes of play, the game awards [tex]\(1 \frac{1}{2}\)[/tex] points.
2. To find the constant of proportionality, we will use the formula:
[tex]\[ \text{constant of proportionality} = \frac{\text{points}}{\text{minutes}} \][/tex]
3. Calculate for the first scenario (4 minutes, [tex]\(\frac{1}{2}\)[/tex] point):
[tex]\[ \text{constant}_1 = \frac{\frac{1}{2}}{4} = \frac{1}{2 \cdot 4} = \frac{1}{8} = 0.125 \][/tex]
4. Calculate for the second scenario (12 minutes, [tex]\(1 \frac{1}{2}\)[/tex] points):
[tex]\[ \text{constant}_2 = \frac{1 \frac{1}{2}}{12} = \frac{\frac{3}{2}}{12} = \frac{3}{2 \cdot 12} = \frac{3}{24} = \frac{1}{8} = 0.125 \][/tex]
5. Both calculations yield the same constant of proportionality, confirming it to be [tex]\(0.125\)[/tex].
Therefore, the constant of proportionality is [tex]\(0.125\)[/tex].
### Part B: Write an equation that represents the relationship
To express the relationship between the number of points [tex]\(y\)[/tex] and the number of minutes [tex]\(x\)[/tex], we use the constant of proportionality [tex]\(k\)[/tex]. The equation will be:
[tex]\[ y = kx \][/tex]
Substituting [tex]\(k = 0.125\)[/tex]:
[tex]\[ y = 0.125x \][/tex]
So, the equation that represents the relationship is:
[tex]\[ y = 0.125x \][/tex]
### Part C: Describe how you would graph the relationship
To graph the relationship between the number of points awarded and the time played, follow these steps:
1. Plot the points representing the scenarios given in the problem:
- (4, 0.5): For 4 minutes of play, 0.5 points are awarded.
- (12, 1.5): For 12 minutes of play, 1.5 points are awarded.
2. Draw a straight line through these points. Since the relationship is linear and proportional, the line should extend through the origin (0,0):
- Point (0,0) is included because playing for 0 minutes results in 0 points.
3. The straight line through (0,0), (4, 0.5), and (12, 1.5) illustrates how the points increase proportionally with the time played.
To describe the graph: "To graph this relationship, plot the points (4, 0.5) and (12, 1.5) on the graph. Draw a straight line through these points, extending through the origin (0,0). This line represents how points increase proportionally with the time played."
### Part D: How many points are awarded for 20 minutes of play?
To find out how many points are awarded for 20 minutes of play, use the equation [tex]\(y = 0.125x\)[/tex] and substitute [tex]\(x = 20\)[/tex]:
[tex]\[ y = 0.125 \cdot 20 = 2.5 \][/tex]
Therefore, for 20 minutes of play, the player is awarded 2.5 points.
### Part A: Find the constant of proportionality
1. From the problem, we know:
- For every 4 minutes of play, the game awards [tex]\(\frac{1}{2}\)[/tex] point.
- For every 12 minutes of play, the game awards [tex]\(1 \frac{1}{2}\)[/tex] points.
2. To find the constant of proportionality, we will use the formula:
[tex]\[ \text{constant of proportionality} = \frac{\text{points}}{\text{minutes}} \][/tex]
3. Calculate for the first scenario (4 minutes, [tex]\(\frac{1}{2}\)[/tex] point):
[tex]\[ \text{constant}_1 = \frac{\frac{1}{2}}{4} = \frac{1}{2 \cdot 4} = \frac{1}{8} = 0.125 \][/tex]
4. Calculate for the second scenario (12 minutes, [tex]\(1 \frac{1}{2}\)[/tex] points):
[tex]\[ \text{constant}_2 = \frac{1 \frac{1}{2}}{12} = \frac{\frac{3}{2}}{12} = \frac{3}{2 \cdot 12} = \frac{3}{24} = \frac{1}{8} = 0.125 \][/tex]
5. Both calculations yield the same constant of proportionality, confirming it to be [tex]\(0.125\)[/tex].
Therefore, the constant of proportionality is [tex]\(0.125\)[/tex].
### Part B: Write an equation that represents the relationship
To express the relationship between the number of points [tex]\(y\)[/tex] and the number of minutes [tex]\(x\)[/tex], we use the constant of proportionality [tex]\(k\)[/tex]. The equation will be:
[tex]\[ y = kx \][/tex]
Substituting [tex]\(k = 0.125\)[/tex]:
[tex]\[ y = 0.125x \][/tex]
So, the equation that represents the relationship is:
[tex]\[ y = 0.125x \][/tex]
### Part C: Describe how you would graph the relationship
To graph the relationship between the number of points awarded and the time played, follow these steps:
1. Plot the points representing the scenarios given in the problem:
- (4, 0.5): For 4 minutes of play, 0.5 points are awarded.
- (12, 1.5): For 12 minutes of play, 1.5 points are awarded.
2. Draw a straight line through these points. Since the relationship is linear and proportional, the line should extend through the origin (0,0):
- Point (0,0) is included because playing for 0 minutes results in 0 points.
3. The straight line through (0,0), (4, 0.5), and (12, 1.5) illustrates how the points increase proportionally with the time played.
To describe the graph: "To graph this relationship, plot the points (4, 0.5) and (12, 1.5) on the graph. Draw a straight line through these points, extending through the origin (0,0). This line represents how points increase proportionally with the time played."
### Part D: How many points are awarded for 20 minutes of play?
To find out how many points are awarded for 20 minutes of play, use the equation [tex]\(y = 0.125x\)[/tex] and substitute [tex]\(x = 20\)[/tex]:
[tex]\[ y = 0.125 \cdot 20 = 2.5 \][/tex]
Therefore, for 20 minutes of play, the player is awarded 2.5 points.
