Problem 69 - Algebra 1

Question

Match the problem with the formula needed to solve the problem. Then use the Guess, Check, and Revise strategy or another problem-solving strategy to solve the problem. Area of a rectangle \(\quad A=l w \quad\) Distance \(\quad d=r t\) Simple interest \(\quad I=P r t \quad\) Volume of a cube Temperature \(\quad C=\frac{5}{9}(F-32)\) Surface area of a cube \(S=6 s^{2}\) What is the average speed of a runner who completes a \(10,000\) -meter race in 25 minutes?

Step-by-Step Solution

Answer

The average speed of the runner is approximately \(6.67\) m/s.

Step 1: Identify the formula

The formula to use in this exercise is the formula for calculating distance \(d = rt\). Since we are calculating for the speed, we rearrange the formula to find \(r = d/t\).

Step 2: Insert the given values

The distance \(d\) in the problem is 10,000 meters and the time \(t\) is 25 minutes. However, speed is generally measured in meters per second (m/s) so we need to convert 25 minutes into seconds which gives \(1500\) seconds. Substituting the given values in the formula we get \(r = 10000/1500\).

Step 3: Calculate the speed

Calculating \(r\) we get \(r = 6.67\) m/s.

Key Concepts

Distance Formula

The distance formula, represented by the equation \(d = r \times t\), is essential for solving a variety of physics and motion problems. Here, \(d\) stands for distance, \(r\) represents the average speed, and \(t\) indicates time. To use this formula, two of the three variables must be known, so that the third can be calculated.

For example, if a runner completes a 10,000-meter race, the distance (\(d\)) is 10,000 meters. If it took them 25 minutes, time (\(t\)) is 25 minutes. To find the average speed (\(r\)), we rearrange the formula to \(r = \frac{d}{t}\). This formula is extremely versatile and can be applied to many real-world problems, from calculating the speed of a vehicle, to figuring out how long it will take a storm to reach a certain point.

Problem-Solving Strategy

A structured approach is often the key to effectively solve mathematical problems. The 'Guess, Check, and Revise' strategy is a part of this structured problem-solving framework. It begins with making an educated guess about the solution, checking if it fits into the given conditions of the problem, and then revising the approach based on feedback.

Using this strategy for our example about the runner: we first guess that to find the average speed we might need to consider the distance and the time taken. We check this by looking at our formulas and see that indeed, \(d = r \times t\) is the formula for distance which correlates with speed and time. If our initial guess was not correct, we would revise our approach until we find the fitting formula. This iterative method of hypothesizing, testing, and refining is not just effective for mathematical problems but is a valuable skill in daily decision-making and scientific inquiries as well.

Unit Conversion

Unit conversion is a critical skill needed for solving problems across various disciplines, including physics, chemistry, and everyday life scenarios. It ensures that all units of measurement within a calculation are compatible.

For instance, in the runner's speed problem, we have the time in minutes but need to convert it into seconds to comply with the standard unit for speed (meters per second). To do this, we multiply the time in minutes by 60, since there are 60 seconds in a minute. Therefore, 25 minutes is equal to \(25 \times 60 = 1500\) seconds. This conversion is imperative to ensure the accuracy of our final answer. Such unit conversions could involve changing centimeters to meters, kilograms to grams, or even more complex conversions like Celsius to Fahrenheit in temperature measurements.