Tyler went to the park to work out. He started with a slow jog at a constant rate for five minutes. He then increased his speed to a faster, constant pace for seven minutes. He took a three-minute break, then ran for another 7.5 minutes at a constant pace to finish his two-mile run. He rested for the remainder of the time. During what domain interval is he at 0.75 miles?

Question

(1 point)
Responses

15≤s≤22.5
15≤s≤22.5

12≤s≤15
12≤s≤15

5≤s<12
5≤s<12

22.5≤s≤25

Answer 1

To determine during what time interval Tyler is at 0.75 miles, let's break down his workout:

1. First segment: Tyler jogs for 5 minutes at a slow, constant rate.

2. Second segment: He then increases his speed for 7 minutes at a faster, constant pace.

3. Break: After these two segments, Tyler takes a 3-minute break.

4. Final segment: He runs for another 7.5 minutes at a constant pace to finish his two-mile run.

To find the time he reaches 0.75 miles, we need to analyze his pace during each segment:

1. First segment (5 minutes): Let's assume he jogs at a certain speed. Since we don't know the exact speed, we'll leave it as "slow," but he doesn't reach 0.75 miles in this segment.

2. Second segment (7 minutes): Assume he speeds up. If we denote his slower speed as \( s_1 \) during the first segment and his faster speed as \( s_2 \) during the second segment, then:

   • Distance from the first 5 minutes: \( d_1 = s_1 \cdot 5 \)
   • Distance from the next 7 minutes: \( d_2 = s_2 \cdot 7 \)

   We want to find when they sum up to at least 0.75 miles.

3. Break (3 minutes): He does not cover any distance here.

4. Final segment (7.5 minutes): He finishes the 2 miles here, but let's see how much distance he covers before this segment starts.

Let's summarize the time in minutes:

• Total time before the break: 5 + 7 = 12 minutes
• Total run time: 12 minutes + 3 minutes break + 7.5 = 22.5 minutes total.

Now, we know the total distances are:

• Assume he has reached 0.75 miles after the second segment but before the break starts.

Time intervals:

1. 0 to 5 minutes: He is not at 0.75 miles.
2. 5 to 12 minutes: He might reach 0.75 miles sometime during this interval depending on his pace.
   • If he runs enough speed in the second segment, he could achieve 0.75 miles by the end of 12 minutes.
3. 12 to 15 minutes: At this point, he's taking a break, so still not at 0.75 miles.
4. After the break until 22.5 minutes: he's running to finish but has reached 0.75 miles before breaking.

Thus, the interval during which Tyler is at 0.75 miles must be in the second segment, i.e., 5≤s<12 minutes.

Therefore, the correct response is 5≤s<12.