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9. Two car washes charge a basic fee plus a fee based on the number of extras that are chosen. The table below shows the total costs for different car washes at Bubbles Car Wash. The total cost [tex]y[/tex] (in dollars) for a car wash with [tex]x[/tex] extras at Soapy Car Wash is represented by the equation [tex]y = x + 9[/tex].

a) Which car wash charges more for the basic fee?
b) How many extras must be chosen for the total costs to be the same?

\begin{tabular}{|l|c|c|c|c|}
\hline
Number of extras, [tex]x[/tex] & 2 & 4 & 6 & 8 \\
\hline
Total cost, [tex]y[/tex] & 9 & 12 & 15 & 18 \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
To solve the given problem, we'll analyze the cost structure of both Bubbles Car Wash and Soapy Car Wash.

### Step-by-Step Solution

#### 1. Determine the cost equation for Bubbles Car Wash.

We are given the table for Bubbles Car Wash:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Number of extras, } x & 2 & 4 & 6 & 8 \\ \hline \text{Total cost, } y & 9 & 12 & 15 & 18 \\ \hline \end{array} \][/tex]

This can be viewed as points [tex]\((x, y)\)[/tex] on a graph. For Bubbles Car Wash, we identify the cost [tex]\(y\)[/tex] as a linear function of the number of extras [tex]\(x\)[/tex]. To find this linear function, we use linear regression to fit a straight line [tex]\(y = mx + b\)[/tex] where [tex]\(m\)[/tex] is the cost per extra and [tex]\(b\)[/tex] is the basic fee.

#### Calculate the slope (m) and intercept (b) of the line:

Given the table data:
- When [tex]\(x = 2\)[/tex], [tex]\(y = 9\)[/tex]
- When [tex]\(x = 4\)[/tex], [tex]\(y = 12\)[/tex]
- When [tex]\(x = 6\)[/tex], [tex]\(y = 15\)[/tex]
- When [tex]\(x = 8\)[/tex], [tex]\(y = 18\)[/tex]

Using these points, the line of best fit yields the coefficients for the linear equation. The slope [tex]\(m\)[/tex] and the intercept [tex]\(b\)[/tex] describe the cost behavior at Bubbles Car Wash.

After calculations:
- The slope [tex]\(m\)[/tex] is approximately 1.5 (which means the cost per extra is [tex]$1.50). - The intercept \(b\) is approximately 6 (which means the basic fee is $[/tex]6.00).

Thus, the cost function for Bubbles Car Wash is:
[tex]\[ y = 1.5x + 6 \][/tex]

#### 2. Examine the cost equation for Soapy Car Wash.

The cost equation for Soapy Car Wash is given as:
[tex]\[ y = x + 9 \][/tex]
This means the cost per extra is [tex]$1.00 and the basic fee is $[/tex]9.00.

#### Compare the basic fees:

- Bubbles Car Wash: [tex]$6.00 - Soapy Car Wash: $[/tex]9.00

Therefore, Soapy Car Wash charges more for the basic fee.

#### 3. Determine the number of extras where total costs are the same.

To find when the total costs for both car washes are the same, set the equations equal to each other:
[tex]\[ 1.5x + 6 = x + 9 \][/tex]

Solve for [tex]\(x\)[/tex]:
[tex]\[ 1.5x + 6 = x + 9 \][/tex]
[tex]\[ 1.5x - x = 9 - 6 \][/tex]
[tex]\[ 0.5x = 3 \][/tex]
[tex]\[ x = \frac{3}{0.5} = 6 \][/tex]

So, [tex]\(x = 6\)[/tex]. Therefore, 6 extras must be chosen for the total costs to be the same at both car washes.

### Conclusion

- Soapy Car Wash charges a higher basic fee of [tex]$9.00 compared to Bubbles Car Wash which charges $[/tex]6.00.
- The number of extras that must be chosen for both car washes to have the same total cost is 6.
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