Question

You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 km/h (65 mi/h), and the trip takes 1 h and 50 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 km/h (43 mi/h). How much longer does the trip take?

Short answer

The trip takes 55 minutes longer on Friday afternoon.

Step-by-step solution

Convert Trip Time to Hours

The trip normally takes 1 hour and 50 minutes. To convert this into hours, consider that 50 minutes is 50/60 hours. So, the time is: $$1+\frac{50}{60}=1.833\text{ hours.}$$

Calculate Normal Trip Distance

Use the formula for distance, $\text{Distance}=\text{Speed}\times \text{Time}$. Given the speed is 105 km/h and the time is 1.833 hours, compute: $$105\text{ km/h}\times 1.833\text{ hours}=192.465\text{ km.}$$

Calculate Time for Slower Speed

When driving at 70 km/h, use the distance calculated in Step 2. Use the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$. Compute the time: $$\frac{192.465\text{ km}}{70\text{ km/h}}=2.75\text{ hours.}$$

Calculate Extra Time

To find the additional travel time, subtract the normal time from the time taken with the slower speed: $$2.75\text{ hours}-1.833\text{ hours}=0.917\text{ hours.}$$

Convert Extra Time to Minutes

Convert the extra time back to minutes since 0.917 hours need to be expressed in minutes: $$0.917\times 60=55\text{ minutes.}$$

Key concepts

Average Speed

Average speed is a key concept in understanding how quickly or slowly a journey takes place. It is defined as the total distance traveled divided by the total time taken. In simple terms, it is how fast you are moving on average over a stretch of your journey.
For example, when driving between San Diego and Los Angeles at an average speed of 105 km/h, your average speed indicates that for every hour you travel, you cover 105 kilometers.
To further simplify:

• Average speed helps in determining your overall rate of motion.
• It smoothens out all variations in speed that might occur during the trip.
• It's crucial for calculating travel time and planning journeys effectively.

Average speed can hugely affect how long a trip takes, just like in the case of heavy traffic decreasing the speed to 70 km/h and thereby increasing travel time.

Distance Calculation

Distance calculation involves determining the length of the path traveled during a journey. It is an essential aspect as it directly impacts the time required to make a trip.
The formula to calculate distance is: $$\text{Distance}=\text{Speed}\times \text{Time}$$ In the example where you travel from San Diego to Los Angeles under normal conditions, the distance was calculated using an average speed of 105 km/h over 1.833 hours. This gives:

• $$105\text{ km/h}\times 1.833\text{ hours}=192.465\text{ km}$$

Understanding distance calculation is crucial because it lays the foundation for computing other travel parameters, such as estimated travel time at various speeds.

Time Conversion

Time conversion is about changing time from one unit to another and is often necessary for calculations involving different units of time. For example, you may need to convert minutes into hours or vice versa to align with speed units like km/h.
In exercises like the freeway trip from San Diego to Los Angeles, time conversion takes place when converting 1 hour and 50 minutes into purely hours by doing:

• Convert 50 minutes into hours as: $$\frac{50}{60}\text{ hours}=0.833\text{ hours}$$
• Added to the initial 1 hour, gives: $$1+0.833=1.833\text{ hours}$$

This process allows for more straightforward calculations when matching against speeds represented in the same units (hours in this case). It's crucial in creating cohesive and correct calculations.

Travel Time Calculation

Travel time calculation allows you to estimate how long a journey will take given a particular speed and distance. Calculating travel time is necessary for planning and adjusting your expectations for travel durations under different conditions.
You use the formula:

• $$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$$

For instance, when calculating how long it takes to travel 192.465 km at a reduced speed of 70 km/h, you plug into the formula to get:

• $$\frac{192.465\text{ km}}{70\text{ km/h}}=2.75\text{ hours}$$

Recognizing how traffic conditions affect speed, and thus travel time can lead to added time on your journey, such as the extra 0.917 hours calculated, which converts to about 55 minutes. Understanding these principles enhances planning and accurate time management.