Problem 32 - Prealgebra

Question

Use the rate equation \(d=r \cdot t\) to solve. At 1: 30 P.M. Cary leaves his house and drives at an average speed of 65 miles per hour to his brother's house. He arrives at 5: 30 P.M. a. How many hours was the drive to his brother's house? b. How many miles from his brother's house does Cary live?

Step-by-Step Solution

Answer

a. 4 hours b. 260 miles

Step 1: Calculate Elapsed Time

To find out how many hours the drive took, calculate the time between 1:30 P.M. and 5:30 P.M. From 1:30 P.M. to 5:30 P.M., there are 4 hours.

Step 2: Apply the Rate Equation

We know the rate equation is \(d = r \cdot t\). Here, \(r = 65\) miles per hour and \(t = 4\) hours. Substitute these values into the equation: \(d = 65 \cdot 4\).

Step 3: Calculate Distance

Perform the multiplication from the equation substituted: \(d = 65 \times 4 = 260\). Thus, the distance is 260 miles.

Key Concepts

Distance Calculation

Distance calculation is essential when trying to find out how far someone has traveled. This is usually done using the equation \(d = r \cdot t\), where \(d\) represents distance, \(r\) is the rate (or speed), and \(t\) is the time. In our scenario, Cary's driving distance is what we're solving for. The rate Cary travels at is given as 65 miles per hour. We already found out from other calculations how long he drove, which is 4 hours. By plugging these numbers into the equation:

• \(r = 65\) mph
• \(t = 4\) hours

The calculation looks like this: \(d = 65 \cdot 4\). Perform this multiplication to get Cary’s traveled distance: 260 miles. This demonstrates how we use the rate equation to find distances in real-world scenarios.

Elapsed Time

Elapsed time is the total time that passes from the start of an activity to its end. It is a straightforward but crucial element when calculating distances or planning trips. For example, if you start driving at one time and stop at another, the elapsed time gives you the duration of the drive. In Cary's journey, he leaves at 1:30 P.M. and arrives at 5:30 P.M. Calculating the elapsed time involves counting the number of hours and minutes between these two points. Here, it is simple because Cary leaves and arrives on the same day:

• Start time: 1:30 P.M.
• End time: 5:30 P.M.
• Elapsed time: 5:30 - 1:30 = 4 hours

This elapsed time is critical as it directly influences how distance is calculated when paired with a rate or speed.

Average Speed

Average speed is a measure of how fast something or someone moves, calculated over a distance and time. It differs from instantaneous speed, which is the speed at an exact moment. Average speed smooths out any variations in speed that occur during travel. In Cary's case, his average speed between point A (home) and point B (his brother’s house) is given as 65 miles per hour. This tells us that throughout the entire 4-hour drive, Cary traveled at a pace that averages to 65 mph.

• Consistent speed assumption: 65 mph throughout the trip.
• This speed helps calculate distance accurately when the time is known, using \(d = r \cdot t\).

Understanding average speed helps predict travel time and assess how efficiently Cary reaches his destination. It also lays the foundation for more complex calculations involving varying speeds.