Answer :
To solve this problem, we need to find the total cost of playing 20 games at the arcade, given that the relationship between the number of games played and the total cost is linear. We have the following data:
- When 6 games are played, the total cost is [tex]$8.
- When 12 games are played, the total cost is $[/tex]11.
Since it's a linear relationship, we can use these points to find the equation of the line in the format [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here's how we can do this step-by-step:
1. Calculate the Slope (Rate of Change):
The slope, [tex]\( m \)[/tex], measures how much the cost changes for each additional game played. It can be calculated as:
[tex]\[
m = \frac{{\text{{Change in Total Cost}}}}{{\text{{Change in Number of Games Played}}}} = \frac{{11 - 8}}{{12 - 6}} = \frac{3}{6} = 0.5
\][/tex]
This means that for each additional game played, the cost increases by [tex]$0.50.
2. Find the Y-intercept:
The y-intercept, \( b \), is the total cost when no games are played. To find it, use one of the points (for example, when 6 games are played) and the slope:
\[
8 = 0.5 \times 6 + b
\]
Solve for \( b \):
\[
8 = 3 + b \quad \Rightarrow \quad b = 8 - 3 = 5
\]
So, the y-intercept is $[/tex]5.00.
3. Write the Equation of the Line:
Now that we have the slope and y-intercept, the equation of the line is:
[tex]\[
\text{Total Cost} = 0.5 \times (\text{Number of Games Played}) + 5
\][/tex]
4. Calculate the Cost for 20 Games:
Substitute 20 for the number of games played in the equation:
[tex]\[
\text{Total Cost} = 0.5 \times 20 + 5 = 10 + 5 = 15
\][/tex]
Therefore, the total cost to play 20 games at the arcade is $15.
- When 6 games are played, the total cost is [tex]$8.
- When 12 games are played, the total cost is $[/tex]11.
Since it's a linear relationship, we can use these points to find the equation of the line in the format [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here's how we can do this step-by-step:
1. Calculate the Slope (Rate of Change):
The slope, [tex]\( m \)[/tex], measures how much the cost changes for each additional game played. It can be calculated as:
[tex]\[
m = \frac{{\text{{Change in Total Cost}}}}{{\text{{Change in Number of Games Played}}}} = \frac{{11 - 8}}{{12 - 6}} = \frac{3}{6} = 0.5
\][/tex]
This means that for each additional game played, the cost increases by [tex]$0.50.
2. Find the Y-intercept:
The y-intercept, \( b \), is the total cost when no games are played. To find it, use one of the points (for example, when 6 games are played) and the slope:
\[
8 = 0.5 \times 6 + b
\]
Solve for \( b \):
\[
8 = 3 + b \quad \Rightarrow \quad b = 8 - 3 = 5
\]
So, the y-intercept is $[/tex]5.00.
3. Write the Equation of the Line:
Now that we have the slope and y-intercept, the equation of the line is:
[tex]\[
\text{Total Cost} = 0.5 \times (\text{Number of Games Played}) + 5
\][/tex]
4. Calculate the Cost for 20 Games:
Substitute 20 for the number of games played in the equation:
[tex]\[
\text{Total Cost} = 0.5 \times 20 + 5 = 10 + 5 = 15
\][/tex]
Therefore, the total cost to play 20 games at the arcade is $15.
