Problem 42 - College Algebra

Question

Distance equals rate times time, \(d=r t\) . Find the distance Tom travels if he is moving at a rate of 55 \(\mathrm{mi} / \mathrm{h}\) for 3.5 \(\mathrm{h}\) .

Step-by-Step Solution

Answer

Tom travels 192.5 miles.

Step 1: Understanding the Problem

We are given the formula for distance as \(d = rt\), where \(d\) is the distance, \(r\) is the rate (speed), and \(t\) is the time. We need to find out how far Tom travels in a given time at a given speed.

Step 2: Identifying Given Information

From the problem, we know that Tom's rate \(r\) is 55 miles per hour and he travels for \(t = 3.5\) hours.

Step 3: Applying the Formula

Substitute the given values of \(r = 55\) and \(t = 3.5\) into the formula \(d = rt\). We calculate: \[d = 55 \times 3.5\]

Step 4: Performing the Calculation

Calculate the expression \(55 \times 3.5\) to find the distance. Multiply: \[55 \times 3.5 = 192.5\]

Step 5: Conclusion

The distance Tom travels is 192.5 miles according to the formula and calculations.

Key Concepts

Rate

In the context of the distance formula, rate refers to how fast something is moving. It's often expressed in units of distance per unit of time, such as miles per hour (mph) or kilometers per hour (km/h).

• The rate is a crucial part of the formula for calculating distance: \(d = rt\), where \(d\) is distance, \(r\) is rate, and \(t\) is time.
• In our exercise, Tom's rate is given as 55 miles per hour. This tells us how many miles Tom travels in one hour.

Rates can vary, meaning they can change over time. However, in our exercise, Tom maintains a constant rate throughout his journey, simplifying our calculations. Always remember, if rate changes, you'll need to adjust your calculations accordingly.

Time

Time is a fundamental factor when using the distance formula. It tells us the duration the object has traveled at the given rate. In this particular exercise, we are told that Tom travels for 3.5 hours.

• Time is usually given in hours when discussing how far something travels. It's important to match the time unit with the rate's time unit to ensure the consistency of units, like miles per hour and hours.

Understanding time in this context helps us determine how long the rate has been applied to calculate the distance. If time or rate units don't match, remember to convert them so that they do. This matching ensures accurate calculations of the total distance covered.

Calculation

The calculation process involves applying the given formula to determine the unknown value. Here, we want to find the distance Tom travels.

• The primary equation we'll use is \(d = rt\), which allows us to calculate distance when both rate and time are known.
• Substituting in Tom's rate (55 miles per hour) and time (3.5 hours), we get: \(d = 55 \times 3.5\)
• By multiplying these two numbers, we execute the core calculation: \(192.5\), which is the distance Tom travels.

Calculations like these are straightforward if we properly plug in our known values into the formula. Ensuring accurate calculations requires careful attention to detail, such as checking units and mathematical operations.