Problem Setup: Catch Thief Scenario

This problem involves calculating the time it takes for the owner to catch the thief, considering their different start times and speeds.

Calculate Thief's Head Start Distance

The thief starts at 3 p.m. and the owner starts at 4 p.m. This means the thief has a 1-hour head start.

• Thief's Speed = $50\text{ km/h}$
• Head Start Duration = $4\text{ p.m.} - 3\text{ p.m.} = 1\text{ hour}$
• Distance covered by thief in 1 hour = Speed $\times$ Time
• Distance = $50\text{ km/h} \times 1\text{ h} = 50\text{ km}$

When the owner starts at 4 p.m., the thief is already 50 km ahead.

Calculate Relative Speed

The owner is chasing the thief. To find how quickly the distance between them closes, we calculate their relative speed.

• Owner's Speed = $70\text{ km/h}$
• Thief's Speed = $50\text{ km/h}$
• Relative Speed = Owner's Speed - Thief's Speed
• Relative Speed = $70\text{ km/h} - 50\text{ km/h} = 20\text{ km/h}$

The distance between the owner and the thief decreases at a rate of 20 km every hour.

Calculate Time to Catch Up

Now, we calculate the time needed for the owner to cover the 50 km head start distance at the relative speed.

• Time = Distance / Relative Speed
• Time = $50\text{ km} / 20\text{ km/h}$
• Time = $2.5\text{ hours}$

Determine the Final Catch-Up Time

The owner starts at 4 p.m. Add the calculated time to catch up to the owner's start time.

• Owner's Start Time = $4:00\text{ p.m.}$
• Time to Catch Up = $2.5\text{ hours} = 2\text{ hours and } 30\text{ minutes}$
• Catch-Up Time = $4:00\text{ p.m.} + 2\text{ hours } 30\text{ minutes}$
• Catch-Up Time = $6:30\text{ p.m.}$