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Sagot :
Sure, let's analyze the given equations for both members and nonmembers of the arcade and compare their respective graphs.
### Members' Cost Equation
The equation for the yearly cost, \( y \), in dollars for a member based on the total game tokens purchased, \( x \), is given by:
[tex]\[ y = \frac{1}{10} x + 60 \][/tex]
This equation is in the form of \( y = mx + b \), where:
- \( m = \frac{1}{10} \): This is the slope of the line, which represents the cost per token.
- \( b = 60 \): This is the y-intercept of the line, which represents the fixed annual membership fee.
### Nonmembers' Cost Equation
The equation for the yearly cost, \( y \), in dollars for a nonmember based on the total game tokens purchased, \( x \), is given by:
[tex]\[ y = \frac{1}{5} x \][/tex]
This equation is also in the form \( y = mx \), where:
- \( m = \frac{1}{5} \): This is the slope of the line, representing the cost per token.
- \( b = 0 \): There is no fixed cost for nonmembers, so the y-intercept is zero.
### Comparing the Graphs
Now, let's analyze how the graphs of these two equations differ:
1. Slopes (Cost per Token):
- The slope for the member's cost equation is \(\frac{1}{10}\), which indicates that members pay $0.10 per token.
- The slope for the nonmember's cost equation is \(\frac{1}{5}\), which indicates that nonmembers pay $0.20 per token.
Since \(\frac{1}{5} (0.20)\) is greater than \(\frac{1}{10} (0.10)\), the slope of the nonmember's graph is steeper. This means nonmembers pay more per token than members.
2. Y-Intercepts (Fixed Costs):
- The y-intercept for the member's cost equation is 60, representing an initial fixed membership fee of $60.
- The y-intercept for the nonmember's cost equation is 0, indicating there is no initial fixed cost for nonmembers.
This implies that members start with a higher initial cost due to the membership fee, but their cost per token is lower.
### Graphical Representation
On a coordinate plane where the x-axis represents the number of tokens \( x \) and the y-axis represents the total yearly cost \( y \):
- The member's graph will start at the point (0, 60) and rise with a slope of \(\frac{1}{10}\).
- The nonmember's graph will start at the point (0, 0) and rise with a steeper slope of \(\frac{1}{5}\).
### Intersection Point
The graphs will intersect at a certain point where the costs for members and nonmembers are equal. This point can be calculated by setting the two equations equal to each other:
[tex]\[ \frac{1}{10} x + 60 = \frac{1}{5} x \][/tex]
By solving this equation, we can find the specific number of tokens where the cost becomes the same for both members and nonmembers.
In summary, the graph for nonmembers will be steeper, indicating a higher incremental cost per token, but will start at the origin (0, 0), reflecting no initial fixed cost. The member’s graph, on the other hand, will have a gentler slope, indicating a lower incremental cost per token but will start at (0, 60), reflecting the fixed membership fee.
### Members' Cost Equation
The equation for the yearly cost, \( y \), in dollars for a member based on the total game tokens purchased, \( x \), is given by:
[tex]\[ y = \frac{1}{10} x + 60 \][/tex]
This equation is in the form of \( y = mx + b \), where:
- \( m = \frac{1}{10} \): This is the slope of the line, which represents the cost per token.
- \( b = 60 \): This is the y-intercept of the line, which represents the fixed annual membership fee.
### Nonmembers' Cost Equation
The equation for the yearly cost, \( y \), in dollars for a nonmember based on the total game tokens purchased, \( x \), is given by:
[tex]\[ y = \frac{1}{5} x \][/tex]
This equation is also in the form \( y = mx \), where:
- \( m = \frac{1}{5} \): This is the slope of the line, representing the cost per token.
- \( b = 0 \): There is no fixed cost for nonmembers, so the y-intercept is zero.
### Comparing the Graphs
Now, let's analyze how the graphs of these two equations differ:
1. Slopes (Cost per Token):
- The slope for the member's cost equation is \(\frac{1}{10}\), which indicates that members pay $0.10 per token.
- The slope for the nonmember's cost equation is \(\frac{1}{5}\), which indicates that nonmembers pay $0.20 per token.
Since \(\frac{1}{5} (0.20)\) is greater than \(\frac{1}{10} (0.10)\), the slope of the nonmember's graph is steeper. This means nonmembers pay more per token than members.
2. Y-Intercepts (Fixed Costs):
- The y-intercept for the member's cost equation is 60, representing an initial fixed membership fee of $60.
- The y-intercept for the nonmember's cost equation is 0, indicating there is no initial fixed cost for nonmembers.
This implies that members start with a higher initial cost due to the membership fee, but their cost per token is lower.
### Graphical Representation
On a coordinate plane where the x-axis represents the number of tokens \( x \) and the y-axis represents the total yearly cost \( y \):
- The member's graph will start at the point (0, 60) and rise with a slope of \(\frac{1}{10}\).
- The nonmember's graph will start at the point (0, 0) and rise with a steeper slope of \(\frac{1}{5}\).
### Intersection Point
The graphs will intersect at a certain point where the costs for members and nonmembers are equal. This point can be calculated by setting the two equations equal to each other:
[tex]\[ \frac{1}{10} x + 60 = \frac{1}{5} x \][/tex]
By solving this equation, we can find the specific number of tokens where the cost becomes the same for both members and nonmembers.
In summary, the graph for nonmembers will be steeper, indicating a higher incremental cost per token, but will start at the origin (0, 0), reflecting no initial fixed cost. The member’s graph, on the other hand, will have a gentler slope, indicating a lower incremental cost per token but will start at (0, 60), reflecting the fixed membership fee.
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