Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
