https://file.aiquickdraw.com/mathbot/file/17272278888097e5hqolw.jpg

Let's break down the problem in the image step by step.

### Given Information:
- **Tyler's painting rate**: 11 feet per minute.
- **Benito's painting rate**: 15 feet per minute.
- **Fence length**: 150 feet.
- Tyler has already painted \(19 \frac{1}{2} = 19.5\) feet of his side before Benito starts painting.

### Part A: 
*How long will it take for the two sides to have an equal number of feet painted? How many feet will be painted on Benito’s side when the two sides have an equal number of feet painted?*

We need to determine when the number of feet painted by both Benito and Tyler are equal. Let \( t \) represent the time in minutes after Benito starts painting.

- Tyler's total painted feet after \( t \) minutes: \( 19.5 + 11t \) (since Tyler had already painted 19.5 feet and is painting 11 feet/min).
- Benito's total painted feet after \( t \) minutes: \( 15t \).

To find the time when their total painted feet are equal, we set up the equation:

\[
19.5 + 11t = 15t
\]

Solving for \( t \):

\[
19.5 = 15t - 11t
\]
\[
19.5 = 4t
\]
\[
t = \frac{19.5}{4} = 4.875 \text{ minutes}.
\]

At this time, the number of feet painted on each side is equal. Now, let's calculate how many feet Benito will have painted in 4.875 minutes:

\[
\text{Feet painted by Benito} = 15 \times 4.875 = 73.125 \text{ feet}.
\]

Thus, it will take 4.875 minutes for both sides to have equal feet painted, and Benito will have painted 73.125 feet by that time.

---

### Part B: 
*Tyler claims that because he started painting first, he will finish painting his side of the fence before Benito finishes his side. Is this true?*

- Total feet Tyler needs to paint: 150 feet.
- Total feet Benito needs to paint: 150 feet.

Let's calculate the time it takes for each to finish:

- **Tyler's time to finish**: Tyler has 150 - 19.5 = 130.5 feet left to paint. At a rate of 11 feet per minute, his time to finish is:

\[
t_{\text{Tyler}} = \frac{130.5}{11} \approx 11.86 \text{ minutes}.
\]

- **Benito's time to finish**: Benito needs to paint all 150 feet. At a rate of 15 feet per minute, his time to finish is:

\[
t_{\text{Benito}} = \frac{150}{15} = 10 \text{ minutes}.
\]

Since \( t_{\text{Benito}} = 10 \text{ minutes} \) is less than \( t_{\text{Tyler}} \approx 11.86 \text{ minutes} \), Tyler's claim is false. Benito will finish first.

---

### Part C:
*The painter who finishes first gets to rest while the other painter finishes. How long will the painter who finishes first get to rest?*

Since Benito finishes first in 10 minutes and Tyler finishes in approximately 11.86 minutes, the time Benito gets to rest is:

\[
\text{Rest time} = 11.86 - 10 = 1.86 \text{ minutes}.
\]

---

Would you like any further details or clarifications? Here are 5 related questions to extend this:

1. How much fence would Benito paint in 1 minute if he increased his speed by 20%?
2. If Tyler increased his painting rate by 2 feet per minute, would he finish before Benito?
3. What if the fence was 200 feet long? How would this change the results for Part A, B, and C?
4. How long would it take if both painters worked together on the same side of the fence?
5. What would the final outcome be if both painters decreased their speeds by 10%?

**Tip:** When solving rate problems, setting up clear equations based on the given rates and known variables can help simplify complex multi-step problems.

Benito and Tyler are painting opposite sides of the same fence. Tyler has already painted 19.5 feet of his side when Benito starts painting. Benito paints at a rate of 15 feet/min and Tyler paints at 11 feet/min. Part A: How long will it take for both sides to have an equal number of feet painted, and how many feet will Benito have painted by then? Part B: Will Tyler finish before Benito because he started earlier? Part C: How long will the painter who finishes first get to rest?

Explore how to solve rate and time problems as Benito and Tyler paint a 150 ft fence. Benito paints at 15 ft/min while Tyler paints at 11 ft/min, having already painted 19.5 ft. Learn how to calculate when they will have painted the same amount, who finishes first, and how long one rests.

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