Problem 63 - Intermediate Algebra

Question

If a car travels at 60 mph for 30 minutes, explain why the distance traveled is not \(60 \cdot 30=1,800\) miles.

Step-by-Step Solution

Answer

The distance is 30 miles, because 30 minutes is 0.5 hours, making the correct calculation \(60 \times 0.5 = 30\) miles.

Step 1: Identify the Units

First, identify the units involved in the problem. The speed of the car is given as 60 miles per hour (mph), and the time traveled is given as 30 minutes.

Step 2: Convert Time to Consistent Units

To ensure unit consistency, convert the time from minutes to hours. Since there are 60 minutes in an hour, 30 minutes is equivalent to 0.5 hours. This conversion is crucial because the speed is measured in miles per hour, not miles per minute.

Step 3: Correct Formula for Distance

Use the correct formula to calculate the distance, which is \( \text{Distance} = \text{Speed} \times \text{Time} \). Thus, the distance traveled is \( 60 \text{ mph} \times 0.5 \text{ hours} \).

Step 4: Calculate the Distance

Calculate the distance using the corrected values: \( 60 \times 0.5 = 30 \). Therefore, the distance traveled by the car is 30 miles.

Key Concepts

Consistent Units

When solving problems involving speed, time, and distance, ensuring consistent units is essential. Imagine trying to solve a problem where the units don't match up—like trying to measure milk in a cup using a yardstick. That's the kind of confusion we aim to avoid by keeping all units consistent.

In our exercise, the speed is given in miles per hour (mph), while time is initially in minutes. To achieve uniformity:

• First, identify all the units involved.
• Convert all measures to matching units.

This approach helps us maintain coherence in calculations and prevent potential errors. In this exercise, time is converted to hours to match the speed's unit of miles per hour.

Distance Formula

Calculating the distance traveled involves a fundamental formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] Using this formula requires us to ensure that both speed and time are expressed in compatible units.

In this particular example:

• The speed is 60 mph.
• The converted time is 0.5 hours.

After ensuring consistent units, plug the values into the distance formula. Here, you calculate the distance as: \[60 \text{ mph} \times 0.5 \text{ hours} = 30 \text{ miles} \] This method demonstrates why incorrect unit conversion could lead to a misunderstanding of the problem, such as initially assuming 1,800 miles.

Time Conversion

Converting time accurately is crucial, especially when it helps align units for calculations like in speed, distance, and time problems.

Time is often given in different formats, such as minutes, hours, or even seconds. Here's how to deal with it:

• Determine the target unit for your calculation—in this case, hours.
• Use basic conversion ratios: for example, 60 minutes equals 1 hour.

For this exercise, we convert 30 minutes into hours using the ratio: \[\frac{30}{60} = 0.5 \text{ hours} \]Accurate conversion ensures compatibility with the speed's unit of miles per hour, leading to a successful distance calculation.