Problem 42 - Elementary Algebra

Question

The distance traveled \(D\) is equal to the average rate \(r\) times the time traveled \(t\) at that rate: \(D=r t .\) Determine the distance traveled given the rate and the time. 75 feet per second for 5 seconds

Step-by-Step Solution

Answer

The distance traveled is 375 feet.

Step 1: Identify the Given Values

First, identify the values given in the problem. The problem states that the rate \(r\) is 75 feet per second and the time \(t\) is 5 seconds.

Step 2: Write Down the Formula

Recall the formula for distance, which is \(D = rt\). This formula tells us how to calculate the distance traveled when we know the rate and the time.

Step 3: Substitute the Values

Replace \(r\) and \(t\) in the formula with the values from the problem: \(r = 75\) feet per second and \(t = 5\) seconds. The formula then becomes \(D = 75 \times 5\).

Step 4: Perform the Calculation

Multiply 75 by 5 to find the distance traveled: \(D = 75 \times 5 = 375\).

Step 5: Interpret the Result

The result of the calculation gives us the distance traveled, which is 375 feet.

Key Concepts

Average Rate

When determining how far an object travels over a certain period, understanding the average rate is crucial. The average rate in mathematics refers to the consistent speed or velocity at which an object travels. It is often expressed as distance per unit time, such as feet per second or miles per hour. In the given exercise, the average rate is provided as 75 feet per second.

This means that every second, the object moves 75 feet. Knowing the average rate allows us to predict how far the object will travel over any given time duration. This simplifies complex travel scenarios by giving us a straightforward measure to calculate expected distances.

Distance Calculation

Distance calculation involves using the relationship between distance, rate, and time. When you know the average rate of travel and the time spent traveling, you can calculate the distance by multiplying them. The formula for this relationship is given as:\[ D = r \, t \]where:

• \(D\) is the distance traveled.
• \(r\) is the rate of travel (average rate).
• \(t\) is the time traveled.

For example, if an object travels at an average rate of 75 feet per second for 5 seconds, you would calculate the distance by substituting these values into the formula:\[ D = 75 \, \text{feet/second} \times 5 \, \text{seconds} \]This results in a total distance of 375 feet. Understanding this calculation lets us solve real-world problems efficiently and accurately.

Time Traveled

Time traveled is a measure of how long an object has been moving. It is a key component in calculating distance when combined with the average rate. In mathematical problems, time can be given in seconds, minutes, hours, etc. It's important to ensure consistency in time units when working with rate units.

For instance, in the exercise, time is given as 5 seconds. This information tells us how long the object has been in motion at the given rate. Knowing the time traveled allows us to apply the distance formula effectively, ensuring we obtain the correct distance traveled in any scenario. Calculating with accurate time units helps avoid mistakes, keeping your computations precise and reliable.