Question

The legal speed limit on a highway is changed from \(55 \mathrm{mi} / \mathrm{h}\) \((=88.5 \mathrm{~km} / \mathrm{h})\) to \(65 \mathrm{mi} / \mathrm{h}(=104.6 \mathrm{~km} / \mathrm{h})\). How much time is thereby saved on a trip from the Buffalo entrance to the New York City exit of the New York State Thruway for someone traveling at the higher speed over this \(435-\mathrm{mi}(=700-\mathrm{km})\) stretch of highway?

Short answer

You save approximately 73 minutes or 1.217 hours when traveling at the higher speed over the 435-mile stretch of highway.

Step-by-step solution

Calculate the Time at Original Speed

Based on the speed formula, rearrange it to time = distance / speed. So, the time (in hours) it will take driving with the original speed (55 mi/h) is \( time_{55} = 435 mi / 55 mi/h = 7.909 \) hours.

Calculate the Time at Higher Speed

Similarly, for the higher speed (65 mi/h), calculate the time with the same formula. So, the time (in hours) it will take to drive with the higher speed is \( time_{65} = 435 mi / 65 mi/h = 6.692 \) hours.

Calculate the Saved Time

Subtract the time at higher speed from time at original speed to find the saved time when driving at higher speed. So, the time saved is \( time_{saved} = time_{55} - time_{65} = 7.909 hours - 6.692 hours = 1.217 hours \). To better understand this figure, convert it into minutes. So, it is \( time_{saved(minutes)} = 1.217 hours * 60 minutes/hour = 73.02 minutes \).

Key concepts

Speed and Distance

Understanding the relationship between speed and distance is essential in solving problems related to travel and motion. Speed is defined as the rate at which an object covers a certain distance. The formula that connects speed, distance, and time is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
This formula can also be rearranged to solve for time, which is particularly useful in travel scenarios where time savings are being calculated:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Using this relationship, if one knows the distance to be traveled and the constant speed of travel, calculating the time it takes to complete the trip becomes straightforward. In the given exercise, by having the distance (435 miles) and two different speeds (55 mph and 65 mph), we can calculate the time taken for each speed. The key here is that speed and distance have a direct relationship. If speed increases while the distance remains constant, the time taken to cover that distance decreases, and vice versa. This phenomenon directly leads to time savings when the speed limit is increased, which is the core principle behind the question in the exercise.

Conversion of Units

Converting units is an indispensable skill in science and everyday life, as different countries and contexts use various measurement systems. In speed and distance calculations, it's common to encounter units like miles (mi), kilometers (km), hours (h), and minutes (min). Being able to convert between these units ensures that comparisons and calculations are accurate and relevant to the context.
For example, in the exercise, speeds are given in miles per hour and equivalent values in kilometers per hour. It is important to understand that
\[ 1 \text{ mi/h} = 1.60934 \text{ km/h} \]
and conversely,
\[ 1 \text{ km/h} = 0.621371 \text{ mi/h} \].
When calculating time in hours and wanting to express it in a more understandable unit like minutes, multiplying by 60 (since there are 60 minutes in 1 hour) accomplishes the conversion:
\[ \text{Time (minutes)} = \text{Time (hours)} \times 60 \]
In the solution, after calculating the saved time in hours, it was converted into minutes for a more tangible understanding of the time saved. These conversions are equally important in painting a clear picture of the total time saved for any given trip.

Subtracting Time Values

When dealing with time values in calculations, we sometimes need to subtract one time value from another. This is especially true in the context of travel time, scheduling, and logistics. To subtract time values, it's important that they are in the same units; otherwise, the result will be incorrect.
In the context of the exercise, we're interested in finding out how much time can be saved by traveling at a higher speed. To do this, we subtract the time it takes to travel at the higher speed from the time it would take to travel at the original speed:
\[ \text{Time Saved} = \text{Time at Original Speed} - \text{Time at Higher Speed} \]
Using this calculation, we find the time saved on the trip when traveling at the higher speed limit. It's important to note that while subtracting larger units of time such as hours, we might want to break them down into smaller units, like minutes, to provide a relatable sense of the duration saved, as was done in the solution provided. This is both beneficial for clear understanding and practical application, such as understanding how much earlier one would arrive at a destination.