Problem 35 - Algebra 1: Concepts and Skills

Question

Find the distance traveled using \(d=r t\). An athlete runs at a rate of 8 feet per second for 5 seconds.

Step-by-Step Solution

Answer

The athlete has traveled a distance of 40 feet.

Step 1: Identify the Values

By reading the problem again, it can be identified that the rate of running, r is equal to 8 feet/second. The time for which the athlete runs, t is equal to 5 seconds.

Step 2: Substitute Values in Formula

The formula by which we can calculate the distance is \(d = r \cdot t\). Substitute the respective values into the formula to get: \(d = 8 \, feet/second \cdot 5 \, seconds\)

Step 3: Calculating the Distance

After substituting the values, multiply the rate and the time to get the distance: \(d = 8 \, feet/second \cdot 5 \, seconds = 40 \, feet\)

Key Concepts

Rate of Change

Understanding the rate of change is crucial when dealing with motion problems. In the context of the exercise, the rate of change represents the speed at which the athlete is running. It is defined as the distance covered per unit of time and is usually expressed in units like feet per second or miles per hour.

In our example, the athlete's rate of change is 8 feet per second, which means for every second that passes, the athlete travels 8 feet. This information is essential because it allows us to calculate how far the athlete will run over a given period by using this constant rate.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and arithmetic operations. In our problem, the algebraic expression we focus on is the distance formula, which is expressed as \(d = r \times t\). Here, \(d\) represents the distance traveled, \(r\) is the rate of change (speed), and \(t\) symbolizes the time.

By substituting the known values into the algebraic expression, we transform an abstract formula into a concrete value that quantifies the athlete's performance. This substitution and simplification of expressions are foundational algebra skills that students need to master.

Unit Conversion

Unit conversion is a necessary skill to solve math problems accurately, especially when dealing with different measurement systems or when measurements are given in various units. In our example, the units for rate and time are consistent (feet per second and seconds), so no conversion is needed.

However, if the time was given in minutes, we would need to convert either speed to feet per minute or time to seconds to keep the units consistent. Remember, the key here is to ensure that units match across the multiplication operation to get a valid result for the distance.

Problem-Solving Strategies

Effective problem-solving strategies are the backbone of mastering mathematical exercises. In this scenario, the strategy involves a step-by-step approach: starting with understanding the problem, identifying the known values, selecting the correct formula, substituting, and then computing.

Always start by carefully reading the exercise to understand what is being asked. Next, pinpoint the variables and their values. Select the appropriate mathematical model or formula that relates the variables. Substitute the known values into the formula carefully, and finally, compute the answer while paying attention to units and arithmetic operations. This structured approach simplifies complex problems and makes them more approachable.