To determine which graph and which equation represent the amount of money Chelsea earns in one day, [tex]\( A \)[/tex], after taking [tex]\( w \)[/tex] walks with the dogs, let's analyze the given information step-by-step.
### Step-by-Step Solution:
1. Understand the Problem:
- Chelsea charges a base rate of [tex]$12 per day.
- For each walk with the dogs, she charges an additional $[/tex]2.50.
2. Formulate the Equation:
- Let [tex]\( A \)[/tex] represent the total amount Chelsea earns in a day.
- Let [tex]\( w \)[/tex] represent the number of walks Chelsea takes.
- The total earnings [tex]\( A \)[/tex] consist of a base charge plus an additional charge per walk.
3. Construct the Earnings Formula:
- Base charge: [tex]$12.
- Additional charge per walk: $[/tex]2.50 \times w.
- Combining these, the total earnings [tex]\( A \)[/tex] can be represented by the equation:
[tex]\[
A = 12 + 2.5w
\][/tex]
4. Match the Equation:
- Compare the resulting equation with the options provided:
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Equations } \\
\hline$A=2.5 w+12$ & $A=2.5+12 w$ \\
\hline$A=(12+2.5) w$ & $A=12-2.5 w$ \\
\hline
\end{tabular}
\][/tex]
- The correct equation is [tex]\( A = 2.5w + 12 \)[/tex].
5. Validate with an Example:
- To ensure correctness, let’s consider an example:
- Assume Chelsea takes 5 walks ([tex]\( w = 5 \)[/tex]).
- Substituting [tex]\( w = 5 \)[/tex] into the equation [tex]\( A = 2.5w + 12 \)[/tex]:
[tex]\[
A = 2.5 \times 5 + 12 = 12.5 + 12 = 24.5
\][/tex]
- Therefore, the total earnings [tex]\( A \)[/tex] is $24.50.
Since the equation [tex]\( A = 2.5w + 12 \)[/tex] accurately represents the scenario described, it is the correct choice. This equation should also correspond with the appropriate graph, which would typically be a straight line indicating Chelsea’s earnings increase linearly with the number of walks.
Thus, the correct equation representing the amount of money Chelsea earns in one day after taking [tex]\( w \)[/tex] walks with the dogs is:
[tex]\[
A = 2.5w + 12
\][/tex]
