Problem 128 - Elementary and Intermediate Algebra

Question

Finding Rates. \(\quad\) A student drove a distance of 135 miles at an average speed of 50 mph. How much faster would she have to drive on the return trip to save 30 minutes of driving time?

Step-by-Step Solution

Answer

She needs to drive 11.36 mph faster on the return trip.

Step 1: Calculate the original time taken

First, we'll calculate how long it took the student to travel 135 miles at a speed of 50 mph. Use the formula for time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). So, \( \text{Time} = \frac{135}{50} = 2.7 \) hours.

Step 2: Determine the desired time for return trip

Next, since she wants to save 30 minutes on the return trip, we need to calculate the time for the return trip. Subtract 0.5 hours (which is 30 minutes) from the original time: \( 2.7 - 0.5 = 2.2 \) hours.

Step 3: Set up the equation for the new speed

To find the needed speed for this new time, we use the same formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). Plugging in the new time, \( \text{Speed} = \frac{135}{2.2} \).

Step 4: Calculate the required speed for the return trip

Calculate the speed using the equation from the previous step: \( \text{Speed} = \frac{135}{2.2} \approx 61.36 \) mph.

Step 5: Find the increase in speed

To determine how much faster she needs to drive, subtract the original speed from this new speed: \( 61.36 - 50 = 11.36 \) mph.

Key Concepts

Average Speed

Understanding average speed is key when solving distance problems related to travel. Average speed is defined as the total distance traveled divided by the total time taken to travel that distance. It gives you a good idea of the overall pace maintained during the trip. In our example, the student traveled 135 miles at an average speed of 50 mph. By using the formula:

• Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} \)

We find that she took 2.7 hours for her trip because 135 divided by 50 equals 2.7. Even though speed can vary during a trip due to traffic or other conditions, the average speed sums it all up in a single convenient number.

Time Calculation

In distance-related problems, calculating time is crucial to planning and optimizing travel. Time tells us how long it takes for a journey and can be calculated using the formula:

• Time = \( \frac{\text{Distance}}{\text{Speed}} \)

Using this formula in our exercise, the student originally took 2.7 hours to travel 135 miles at 50 mph. If she wants to cut down her return time by 30 minutes, she has to adjust her speed. We convert those 30 minutes into hours (0.5 hours) and subtract them from her original time. So, for her return trip, the target time becomes 2.2 hours. Understanding how to manipulate time with these simple conversions is helpful for solving real-world problems efficiently.

Rate of Speed

The rate of speed essentially describes how fast someone or something is moving. It is defined as the distance traveled per unit time. Speed is often expressed in units such as miles per hour or kilometers per hour. In our problem, we need to determine a new rate of speed for the student's return trip to save half an hour. We use the formula:

• Speed = \( \frac{\text{Distance}}{\text{Time}} \)

Plugging in the values for her return trip, we get Speed = \( \frac{135}{2.2} \), which results in approximately 61.36 mph. To find out how much faster the student needs to drive, we take the difference between this new speed and the original speed of 50 mph. Therefore, she needs to increase her speed by roughly 11.36 mph to achieve her goal. Rate of speed adjustments like this are common when trying to meet specific travel deadlines or schedules.