Let's analyze the problem step-by-step to determine who had a head start and how many miles was the head start.
### Step 1: Understand the Problem
Rita and Roger are driving at constant speeds from Phoenix to San Diego. We have data for Rita's distances at specific times, but Roger's distances are not given. We need to determine if there is a head start and the distance of the head start.
### Step 2: Record Rita's Distances
We know Rita's distances at different times:
- At 1 hour: 65 miles
- At 2 hours: 130 miles
- At 3 hours: 195 miles
- At 4 hours: 260 miles
### Step 3: Calculate Rita's Speed
Since Rita travels at a constant speed, we can calculate her speed using the given data. Using any point of data, for example:
- At 1 hour, Rita traveled 65 miles.
Speed of Rita = Distance / Time = 65 miles / 1 hour = 65 miles per hour.
So Rita travels at a constant speed of 65 miles per hour.
### Step 4: Assume Roger's Speed
For the sake of this problem, let's assume that Roger also travels at a constant speed and that his speed is the same as Rita's (65 miles per hour).
### Step 5: Calculate Roger's Distances
Since Roger travels at the same speed as Rita (65 miles per hour), we can calculate his distances:
- At 1 hour: 65 miles/hour 1 hour = 65 miles
- At 2 hours: 65 miles/hour 2 hours = 130 miles
- At 3 hours: 65 miles/hour 3 hours = 195 miles
- At 4 hours: 65 miles/hour 4 hours = 260 miles
### Step 6: Compare Distances and Determine Head Start
To determine if there was a head start, we compare Rita's distances to the assumed distances of Roger. If Rita's distances at any hour are greater than Roger's calculated distances, it would suggest a head start.
Let's compare:
- At 1 hour:
- Rita: 65 miles
- Roger: 65 miles
- At 2 hours:
- Rita: 130 miles
- Roger: 130 miles
- At 3 hours:
- Rita: 195 miles
- Roger: 195 miles
- At 4 hours:
- Rita: 260 miles
- Roger: 260 miles
### Conclusion
From the comparisons above, we can see that at every given time, Rita and Roger have traveled the same distance. This indicates that neither had a head start. Therefore, both started at the same time, and there was no head start.
### Final Answer
Who had a head start? Neither.
The head start distance? 0 miles.
