Problem 39 - Algebra A Combined Function

Question

The Cat is a high-speed catamaran auto ferry that operates between Bar Harbor, Maine, and Yarmouth, Nova Scotia. The Cat can make the trip in about \(2 \frac{1}{2}\) hours at a speed of \(55 \mathrm{mph}\). About how far apart are Bar Harbor and Yarmouth?

Step-by-Step Solution

Answer

The distance is approximately 137.5 miles.

Step 1: Identify the Formula

To find the distance between two points given time and speed, we use the formula \( \text{Distance} = \text{Speed} \times \text{Time}. \)

Step 2: Convert Time to Miles Per Hour

Convert the time from mixed fractions to an improper fraction: \[2 \frac{1}{2} \text{ hours} = 2.5 \text{ hours}.\]

Step 3: Calculate the Distance

Multiply the speed by the time to find the distance: \[\text{Distance} = 55 \text{ mph} \times 2.5 \text{ hours} = 137.5 \text{ miles}.\]

Step 4: Conclusion

The distance between Bar Harbor and Yarmouth is approximately 137.5 miles.

Key Concepts

Speed

Speed is a measure of how fast something moves over a distance. It is usually calculated as the ratio of distance covered per unit of time. In most cases, the unit of speed is miles per hour (mph) or kilometers per hour (km/h) depending on the region.

Understanding speed is essential for solving problems related to distance calculations. When given the speed of a vehicle, you can determine how long it will take to travel a certain distance or, as in the case of our problem, how far it can go in a given time.

To calculate speed, you use the formula:

• Speed = Distance / Time

In our problem, the speed given is 55 mph, which means the Catamaran travels 55 miles for each hour it is moving.

Time Conversion

Time conversion is crucial when you're working with problems that involve time in formats other than whole numbers. Time can sometimes be represented as mixed numbers, fractions, or decimals. This is common in real-life scenarios where events or actions don't last exactly an hour, half an hour, or any other round number.

To tackle these problems, you need to convert time measurements to a single uniform unit. It simplifies calculations, especially when mixing different units like hours and minutes.

In the example, we convert 2 1/2 hours to a decimal to make the multiplication straightforward.

• 2 1/2 hours is equivalent to 2.5 hours.

Performing this conversion allows us to easily multiply time by speed to find distance.

Algebraic Formula

Algebraic formulas are like recipes; use them for specific outputs when given certain inputs. In the context of distance calculation, you'll often see the formula:

• Distance = Speed × Time

This formula helps in determining one quantity as long as the other two are known. It is quite straightforward because it involves basic multiplication.

Applying formulas ensures that all necessary steps are consistently followed. It also allows tackling similar problems without confusion. In this exercise, multiplying the speed of 55 mph by the time of 2.5 hours directly yielded the distance of 137.5 miles.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They frequently occur in various everyday situations, including time measurements. To handle calculations involving mixed numbers, it's often helpful to convert them to improper fractions or decimals.

For instance, in the current exercise, the travel time of 2 1/2 hours is a mixed number. Converting this mixed number first into a decimal (2.5) is vital for easily multiplying it against the speed (in mph). This conversion avoids any confusion and potential calculation errors.

Understanding how to work with mixed numbers is essential for accurately solving many mathematical problems, particularly in scenarios involving time, rates, or measurements.