Problem 33 - Prealgebra

Question

Use the rate equation \(r=\frac{d}{t}\) to solve. At 2: 30 P.M. Brittney leaves her house and drives 260 miles to her sister's house. She arrives at 6: 30 p.m. a. How many hours was the drive to her sister's house? b. What was Brittney's average speed?

Step-by-Step Solution

Answer

Brittney drove for 4 hours at an average speed of 65 mph.

Step 1: Calculate Duration of the Drive

Brittney leaves at 2:30 PM and arrives at 6:30 PM. To find the duration, subtract the departure time from the arrival time: \(6:30 PM - 2:30 PM = 4\) hours. So, the drive was 4 hours long.

Step 2: Apply the Rate Equation

We are given the distance \(d = 260\) miles. The time \(t\) has been calculated as 4 hours in Step 1. The rate equation is \(r = \frac{d}{t}\). Substitute in the known values: \(r = \frac{260}{4}\).

Step 3: Calculate the Average Speed

Perform the division from Step 2 to find \(r\): \(r = \frac{260}{4} = 65\). So, Brittney's average speed was 65 miles per hour.

Key Concepts

Average Speed Calculation

Understanding how to calculate average speed is fundamental in solving problems involving movement and travel. The average speed of an object can be found using the rate equation: \( r = \frac{d}{t} \), where \( r \) is the rate or average speed, \( d \) is the distance traveled, and \( t \) is the time taken for the journey. In this equation:

• The numerator (distance \( d \)) tells us how far the object has moved.
• The denominator (time \( t \)) indicates how long it took to cover that distance.

By using the rate equation, you can determine the average speed by dividing the total distance by the total time. In the context of Brittney's journey, she traveled a distance of 260 miles over a period of 4 hours. Therefore, her average speed was calculated to be 65 miles per hour. This means that if she traveled at a constant speed, she would cover 65 miles in about every hour of her journey. It's important to note that average speed doesn't account for variations in speed during the trip.

Time Duration

Calculating the duration of a trip involves determining the amount of time taken from the beginning to the end of the journey. This is crucial for understanding how long an activity lasts. To find the time duration, simply subtract the start time from the end time. In Brittney's case:
- She began her trip at 2:30 PM and ended at 6:30 PM. - Subtracting the start time from the end time gives: \(6:30 \ \text{PM} - 2:30 \ \text{PM} = 4 \ \text{hours}\).
This calculation tells us that her drive lasted for a total of 4 hours. Make sure to keep track of AM and PM when subtracting times to avoid confusion. Properly calculating time duration is critical in applying it to other calculations, like determining the average speed.

Distance Calculation

Distance calculation is the process of finding out how far something has traveled. In problems involving moving objects, it's essential as it forms a core part of the rate equation. The formula in rate problems, \( r = \frac{d}{t} \), can be rearranged to find distance: \( d = r \times t \).

• This rearrangement allows calculation of distance if the speed and time are known.
• Knowing the distance helps in visualizing the scope of a journey or travel plan.

In Brittney's scenario, the 260 miles represent the total distance she covered during her drive to her sister's house. Observing the distance in real-world terms helps relate mathematical concepts to actual travel. Distance, together with speed and time, forms the triad that governs motion and movement-related calculations.