To determine the track length given that Michael is 1.5 times as fast as Bob and that Michael gives Bob a head start of 150 meters, we'll need to follow a logical sequence of steps.
1. Define Variables:
- Speed of Bob = \( v \) meters per unit time
- Speed of Michael = \( 1.5v \) meters per unit time
2. Determine Time Taken:
- Let \( t \) be the time it takes for both Michael and Bob to complete the race.
3. Calculate Distances Traveled:
- Distance traveled by Bob in time \( t \):
[tex]\[
\text{Distance by Bob} = v \times t
\][/tex]
- Distance traveled by Michael in the same time \( t \):
[tex]\[
\text{Distance by Michael} = 1.5v \times t
\][/tex]
4. Account for Head Start:
- Since Michael gives Bob a head start of 150 meters, the total distance covered by Michael includes this head start in addition to the track length covered by Bob.
- For the race to end in a tie, the distance covered by Michael must equal the head start plus the distance covered by Bob.
Therefore, the track length \( L \) must be equal to the distance Bob travels plus the head start.
5. Establishing the Equality and Solving:
- Let \( L \) be the track length.
- The distance covered by Bob is \( v \times t \).
- The total distance covered by Michael, including the head start, must equal the track length \( L \).
[tex]\[
1.5v \times t = L
\][/tex]
- Since Michael must also cover the head start distance, relate this to Bob’s travel distance plus head start:
[tex]\[
v \times t + 150 = L
\][/tex]
6. Solve for the Track Length \( L \):
- Both \( 1.5v \times t \) and \( v \times t + 150 \) represent the total distance (track length):
[tex]\[
1.5v \times t = v \times t + 150
\][/tex]
- Let’s isolate and solve for \( L = v \times t + 150 \):
- If we plug in values for easier computation, say \( v = 100 \) (Bob's speed) for a chosen unit time \( t = 1 \) to simplify:
[tex]\[
\text{Distance by Bob} = 100 \times 1 = 100 \text{ meters}
\][/tex]
[tex]\[
L = 100 + 150 = 250 \text{ meters} (track length)
\][/tex]
Thus, the track length [tex]\( L \)[/tex] is 250 meters.
