Answer:
THE ANSWER IS GIVEN BELOW
Step-by-step explanation:
To create a piecewise function describing Rosa's spending pattern, we need to consider the different spending amounts for each time interval. Let's break down the problem step by step:
1. For the first 3 days, Rosa spends $20 each day.
2. For the next 2 days, she spends nothing.
3. After those 5 days, she spends $10 each day until her savings are depleted.
Let's denote \(d\) as the number of days since the start of her vacation.
1. For the first part of the function (1st equation), representing the first 3 days where she spends $20 each day:
\[ f(d) = \begin{cases}
\$20d & \text{if } 0 \leq d \leq 3 \\
\end{cases} \]
2. For the second part of the function (2nd equation), representing the next 2 days where she spends nothing:
\[ f(d) = \begin{cases}
\$20d & \text{if } 0 \leq d \leq 3 \\
0 & \text{if } 3 < d \leq 5 \\
\end{cases} \]
3. For the third part of the function (3rd equation), representing the remaining days where she spends $10 each day until her savings are depleted:
Since Rosa initially saved $100 and spent $20 each day for the first 3 days and spent nothing for the next 2 days, the total amount spent during the first 5 days is \(20 \times 3 + 0 \times 2 = 60\). Therefore, after the first 5 days, Rosa has $100 - $60 = $40 remaining.
Now, for the remaining days where she spends $10 each day until her savings are depleted, we can represent this part of the function as follows:
\[ f(d) = \begin{cases}
\$20d & \text{if } 0 \leq d \leq 3 \\
0 & \text{if } 3 < d \leq 5 \\
\$40 - 10(d - 5) & \text{if } d > 5 \\
\end{cases} \]
This piecewise function represents Rosa's spending pattern during her vacation.
