Let AB be a diameter of a circular track. Tom starts from point A running clockwise on the track. At the same time Jerry starts from Point B running counterclockwise on the track. Each person runs at a constant speed. They first meet after Tom has run 200 feet. They next meet after Jerry has run 300 feet past their first meeting point. What is the length of the track in feet?
1000 ft
because...
200*2=400
and
300*2=600
then
400+600=1000ft
=]
To solve this problem, let's break it down step by step:
Step 1: Draw a diagram and label the given information.
- Draw a circle to represent the circular track.
- Label the diameter AB.
- Remember that Tom runs clockwise and starts from point A.
- Jerry runs counterclockwise and starts from point B.
- Mark the first meeting point as M.
- Indicate that Tom has run 200 feet before the first meeting and Jerry has run 300 feet past their first meeting.
Step 2: Understand the concept of relative speed.
- The relative speed between Tom and Jerry is the sum of their individual speeds because they are running towards each other.
- Let's say Tom's speed is T and Jerry's speed is J.
- The relative speed will be T + J.
Step 3: Determine the time taken for Tom and Jerry to meet.
- The distance Tom runs before the first meeting is 200 feet, which is (1/2) of the total track length.
- The distance Jerry runs past the first meeting is 300 feet.
- Since their speeds are constant, the time taken by Tom and Jerry to meet will be the same.
- Let's assume the time taken by Tom and Jerry to meet is t.
Step 4: Calculate the length of the track using the time and relative speed.
- For Tom, the distance covered in time t is t * T.
- For Jerry, the distance covered in time t is t * J.
- The sum of their distances should equal the total length of the track, which is the circumference of the circle.
- The formula for calculating the circumference of a circle is 2πr, where r is the radius.
- Since AB is the diameter, the radius is (AB/2).
Step 5: Set up equations and solve for the track length.
- From Step 4, we know that t * T + t * J = 2πr.
- Plugging in the values, we have t * T + t * J = 2π(AB/2).
- Simplifying the equation, we get t(T + J) = πAB.
Step 6: Solve for the track length.
- We are given that Tom's distance covered before the first meeting is 200 feet. Since speed = distance/time, we have T = 200/t.
- We are also given that Jerry's distance covered past the first meeting is 300 feet. So J = 300/t.
- Substituting these values into our equation from Step 5, we get t(200/t + 300/t) = πAB.
- Simplifying further, we have 500 = πAB.
- Dividing both sides by π, we get AB = 500/π.
Therefore, the length of the track is 500/π feet.