Sometimes, we know construction or an accident will slow down our travel on our route. A detour might be tempting. We can do some quick math with some estimations in order to determine whether a detour is worth the trouble.
1. Your usual route on the interstate is ten miles on a 65 mile-per-hour highway, then two miles on a 25 mile-per-hour street. The stretch of highway is being worked on all summer and speeds are reduced to 45 miles per hour for the entire ten miles.
You could take an alternate route to work. The alternate route is 55 miles per hour for 11 miles, then 30 miles per hour for 3 miles. Which route is quickest?
The relationship between distance traveled, rate, and time can be expressed with this equation:
\[D = RT\]
We can solve for time in this formula by dividing R from both sides.
\[\frac{D}{R} = T\]
For the first route: The time spent driving at the reduced speed of 45 miles per hour is
\[T = \frac{10 \textbf{ miles}}{45 \textbf{ miles per hour}} \approx 0.222 \textbf{ hours}\]
We can convert hours to minutes.
\[0.222 \textbf{ hours} \cdot \frac{60 \textbf{ minutes}}{1 \textbf{ hour}} =\]
\[0.222 \cancel{\textbf{ hours}} \cdot \frac{60 \textbf{ minutes}}{1 \cancel{\textbf{ hour}}} =\]
13.33 minutes
Two miles at 25 miles per hour means we have a travel time of
\[\frac{2}{25} 0.08 \textbf{ hours}\]
which is \(0.08 \cdot 60 = 4.8\) minutes.
The total travel time on the usual route with delays is 13.33 minutes + 4.8 minutes = 18.13 minutes.
The alternate route is 55 miles per hour for 11 miles, then 30 miles per hour for three miles.
The time spent traveling at 55 miles per hour is
\[\frac{11}{55} = 0.2 \textbf{ hours} = 12 \textbf{ minutes}\]
The time spent traveling at 30 miles per hour is
\[\frac{3}{30} = 0.10 \textbf{ hours} = 6 \textbf{ minutes}\]
The total travel time for the alternate route is 18 minutes. The difference in time between the two routes is almost negligible.
1. A person knows that construction is happening on two different routes that they can take to work. Their map software is not accurately reflecting the delays. They want to do some quick math to determine which route is fastest.
Route A would be approximately 40 miles per hour for 10 miles, then 25 miles per hour for six miles.
Route B would be approximately 30 miles per hour for 8 miles, then 20 miles per hour (due to traffic) for three miles.
Which route is fastest?