21 Categorization 1 point On several occasions, Kurt recorded the number of people already in line when he arrived at a local deli and the amount of time elapsed between when he left for his lunch break and when he returned to his office. Using the data, Kurt made a scatter plot and found the equation of the line of best fit to be $$ y=0.5 x+34.5 $$, where x is the number of people already in line, and $$ y $$ is the elapsed time in minutes. If Kurt's lunch break is 45 minutes, would each of these numbers of people in line result in him returning to work late? Put in the category of Yes or No for each. Yes No Answers Chosen

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To determine whether Kurt will return to work late based on the number of people already in line at the deli, we'll analyze the given linear equation and compare the elapsed time to his lunch break duration.

**Given:**
- **Equation of the line of best fit:** $$ y = 0.5x + 34.5 $$
  - $$ x $$: Number of people already in line
  - $$ y $$: Elapsed time in minutes
- **Lunch break duration:** 45 minutes

**Objective:**
Determine for each number of people ($$ x $$) whether Kurt will return late (Yes) or not (No).

**Analysis:**

1. **Set the elapsed time equal to the lunch break duration to find the cutoff point:**
   $$
   0.5x + 34.5 = 45
   $$
   
2. **Solve for $$ x $$:**
   $$
   0.5x = 45 - 34.5 \
   0.5x = 10.5 \
   x = \frac{10.5}{0.5} \
   x = 21
   $$
   
3. **Interpretation:**
   - **If $$ x > 21 $$:**
     - Elapsed time $$ y = 0.5x + 34.5 > 45 $$ minutes
     - **Kurt returns late.** (Yes)
     
   - **If $$ x \leq 21 $$:**
     - Elapsed time $$ y = 0.5x + 34.5 \leq 45 $$ minutes
     - **Kurt does not return late.** (No)

**Summary:**

| Number of People in Line ($$ x $$) | Returns Late? |
|------------------------------------|----------------|
| $$ x > 21 $$                       | Yes            |
| $$ x \leq 21 $$                    | No             |

**Example:**
- **If there are 25 people in line:**
  $$
  y = 0.5(25) + 34.5 = 12.5 + 34.5 = 47 \text{ minutes}
  $$
  - Since 47 > 45, **Yes**, Kurt returns late.
  
- **If there are 18 people in line:**
  $$
  y = 0.5(18) + 34.5 = 9 + 34.5 = 43.5 \text{ minutes}
  $$
  - Since 43.5 ≤ 45, **No**, Kurt does not return late.

**Conclusion:**
- **Yes:** When the number of people in line is greater than 21.
- **No:** When the number of people in line is 21 or fewer.
| Number of People in Line | Returns Late? | |-------------------------|---------------| | >21 | Yes | | ≤21 | No |

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