A runner runs around a circular track. he completes one lap at a time of t = 490 s at a constant speed of v = 3.7 m/s.
The Correct Answer and Explanation is:
To answer this question, we need to find the radius or the length of the circular track based on the given information.
Given:
- Time to complete one lap, ( t = 490 \, \text{s} )
- Constant speed, ( v = 3.7 \, \text{m/s} )
The distance traveled by the runner for one lap is the circumference of the circular track, ( C ). The formula for the circumference of a circle is:
[
C = 2\pi r
]
where ( r ) is the radius of the circular track.
We can find the circumference using the relationship between distance, speed, and time. The runner’s speed is constant, so the distance traveled in one lap is the product of the speed and the time taken to complete the lap. Therefore, the circumference of the track is:
[
C = v \times t
]
Substituting the given values:
[
C = (3.7 \, \text{m/s}) \times (490 \, \text{s}) = 1813 \, \text{m}
]
Now that we know the circumference, we can solve for the radius of the track. Using the circumference formula:
[
C = 2\pi r
]
[
1813 \, \text{m} = 2\pi r
]
Solving for ( r ):
[
r = \frac{1813}{2\pi} \approx \frac{1813}{6.2832} \approx 288.5 \, \text{m}
]
Final Answer:
The radius of the circular track is approximately 288.5 meters.
Explanation:
The problem provides the time taken to complete one lap and the runner’s constant speed. From this, we calculated the distance traveled in one lap (the circumference), which is directly related to the radius of the circle. By rearranging the formula for the circumference of a circle, we were able to determine the radius, showing how the speed and time information allow us to calculate geometric properties of the track.