Answer :
To determine how long it takes for the barnacle to get back to its starting point, we need to consider the path and speed of the boat.
1. Understanding the Path:
- Assume the boat is traveling in a circular path. The length of the path for one full rotation (one complete circle) is the circumference of the circle.
- The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\(r\)[/tex] is the radius of the circle. However, the radius [tex]\(r\)[/tex] value is not necessary here, as it will simplify out.
2. Understanding the Speed:
- The boat is traveling at a constant speed of 1 meter per second.
3. Calculating the Time:
- The time [tex]\(T\)[/tex] it takes for the barnacle to complete one full rotation around the circle is the circumference divided by the speed:
[tex]\[ T = \frac{C}{v} \][/tex]
where [tex]\(v\)[/tex] is the speed of the boat.
- Since the boat travels 1 meter per second, [tex]\(v = 1\)[/tex].
4. Substituting Known Values:
- The circumference [tex]\(C\)[/tex] is [tex]\(2 \pi \)[/tex] because in the unit circle, [tex]\(r\)[/tex] is 1 meter. Thus,
[tex]\[ T = \frac{2 \pi}{1} = 2 \pi \][/tex]
So, the time it takes for the barnacle to return to its starting point is:
[tex]\[ 2 \pi \text{ seconds} \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ \boxed{2 \pi \text{ seconds}} \][/tex]
1. Understanding the Path:
- Assume the boat is traveling in a circular path. The length of the path for one full rotation (one complete circle) is the circumference of the circle.
- The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\(r\)[/tex] is the radius of the circle. However, the radius [tex]\(r\)[/tex] value is not necessary here, as it will simplify out.
2. Understanding the Speed:
- The boat is traveling at a constant speed of 1 meter per second.
3. Calculating the Time:
- The time [tex]\(T\)[/tex] it takes for the barnacle to complete one full rotation around the circle is the circumference divided by the speed:
[tex]\[ T = \frac{C}{v} \][/tex]
where [tex]\(v\)[/tex] is the speed of the boat.
- Since the boat travels 1 meter per second, [tex]\(v = 1\)[/tex].
4. Substituting Known Values:
- The circumference [tex]\(C\)[/tex] is [tex]\(2 \pi \)[/tex] because in the unit circle, [tex]\(r\)[/tex] is 1 meter. Thus,
[tex]\[ T = \frac{2 \pi}{1} = 2 \pi \][/tex]
So, the time it takes for the barnacle to return to its starting point is:
[tex]\[ 2 \pi \text{ seconds} \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ \boxed{2 \pi \text{ seconds}} \][/tex]
