Finish the two parts of the piecewise function:

Rosa saved $100 to spend on vacation. For the first 3 days of her vacation she spent $20 each day. Then for the next 2 days, she spent nothing. After those 5 days, she spent $10 each day until her savings were depleted.

View photo, I've done a lot of the work, just can't figure out the 1st and 3rd equations.

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:

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2. For the second part of the function (2nd equation), representing the next 2 days where she spends nothing:

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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.