Certainly! Let's solve the problem step-by-step.
1. Parameters Given:
- Radius of the circular track, [tex]\( r = 84 \)[/tex] meters.
- Time taken to complete one round, [tex]\( T = 50 \)[/tex] seconds.
- Total time given, [tex]\( t = 1 \)[/tex] minute and [tex]\( 40 \)[/tex] seconds.
First, convert the total time into seconds:
[tex]\[
t = 1 \times 60 + 40 = 100 \text{ seconds}
\][/tex]
2. Calculate the Number of Rounds Completed:
Determine how many complete rounds the athlete finishes in the given total time.
[tex]\[
\text{Number of rounds completed} = \left\lfloor \frac{t}{T} \right\rfloor = \left\lfloor \frac{100}{50} \right\rfloor = 2
\][/tex]
3. Calculate the Circumference of the Circular Track:
The distance covered in one complete round of the circle is the circumference of the circle, which can be computed using the formula:
[tex]\[
\text{Circumference} = 2 \pi r = 2 \pi \times 84 \approx 527.79 \text{ meters}
\][/tex]
4. Calculate the Total Distance Covered:
Since the athlete completes 2 rounds, the total distance covered is:
[tex]\[
\text{Total distance covered} = \text{Number of rounds completed} \times \text{Circumference} = 2 \times 527.79 \approx 1055.58 \text{ meters}
\][/tex]
5. Calculate the Displacement:
Displacement is the shortest distance from the starting point to the ending point of the motion. After completing full circles, the displacement returns to zero if the start and end points are the same. Here, the athlete finishes exactly 2 full rounds, returning to the starting point.
Therefore, the displacement is:
[tex]\[
\text{Displacement} = 0 \text{ meters}
\][/tex]
### Summary:
- Distance Covered: The athlete covers a total distance of approximately 1055.58 meters after 1 minute and 40 seconds.
- Displacement: The displacement of the athlete is 0 meters since they return to the starting point after completing 2 full rounds.
This completes the detailed step-by-step solution for the given problem.
