After robbing a bank in Dodge City, a robber gallops off at 12 mi/h. 30 minutes later, the marshall leaves to pursue the robber at 16 mi/h.
How long (in hours) does it take the marshall to catch up to the robber?
let the time taken by the robber when he is caught be t hrs
distance he covered = 12t miles
time taken by the marshall to catch him = t - 1/3
distance traveled by him = 16(t - 1/2)
but 12t = 16(t - 1/2)
12t = 16t - 16/2
-3t = -16/2
t = 8/3 hrs
To find out how long it takes the marshall to catch up to the robber, we need to determine the distance the robber travels in that time period.
The robber gallops at 12 mi/h and started 30 minutes earlier than the marshall. So, the robber rides for 30 minutes (which is 30/60 = 0.5 hours) longer than the marshall.
Let's calculate the distance the robber travels in that time period:
Distance = Speed x Time
Distance = 12 mi/h x 0.5 h (30 minutes, converted to hours)
Distance = 6 miles
Now, let's find out how long it takes the marshall to cover this distance at a speed of 16 mi/h:
Time = Distance / Speed
Time = 6 miles / 16 mi/h
Time = 0.375 hours (or 22.5 minutes)
Therefore, it takes the marshall 0.375 hours (or 22.5 minutes) to catch up to the robber.
To find the time it takes for the marshall to catch up to the robber, we need to determine the distance the robber travels before being caught.
Let's break down the problem:
1. The robber travels at a speed of 12 mi/h.
2. The marshall starts chasing 30 minutes (or 0.5 hours) later, at a speed of 16 mi/h.
The distance the robber travels in the 30 minutes before the marshall starts chasing can be calculated by:
Distance = Speed x Time = 12 mi/h x 0.5 h = 6 miles.
Now, let's calculate the time it takes for the marshall to catch up to the robber:
The marshall's speed relative to the robber is the difference between their speeds: 16 mi/h - 12 mi/h = 4 mi/h.
Since they cover the same distance when the marshall catches the robber, we can set up the equation:
Distance = Speed x Time.
Plugging in the values we know:
6 miles (the distance the robber traveled before the marshall started) = 4 mi/h (the relative speed) x Time.
Simplifying the equation:
Time = Distance / Speed = 6 miles / 4 mi/h = 1.5 hours.
Therefore, it takes the marshall 1.5 hours to catch up to the robber.