Problem 67 - Algebra 1

Question

Use the following information.Find a partner to help you with the following experiment. Take a ruler and drop it from a fixed height. Once the ruler is released, your partner should catch it as quickly as possible. Measure how far the ruler falls before your partner catches it. The reaction time can be calculated by the equation $$ t=\sqrt{\frac{d}{192}} $$ where \(t\) is the reaction time (in seconds) and \(d\) is the distance (in inches) the ruler falls. Suppose for three trials, the ruler falls 4 inches, 1 inch, and 3 inches. Calculate the average reaction time for the three trials.

Step-by-Step Solution

Answer

The average reaction time for the three trials is 0.06412 seconds.

Step 1: Apply the given formula for each trial

In the first trial, the ruler falls 4 inches, in the second trial it falls 1 inch and in third it falls 3 inches. The reaction times can be calculated via the formula \( t=\sqrt{\frac{d}{192}} \), where \(d\) is the respective distance the ruler fell. So we calculate the three reaction times as follows: \( t_1 =\sqrt{\frac{4}{192}} \), \( t_2 =\sqrt{\frac{1}{192}} \), and \( t_3 =\sqrt{\frac{3}{192}} \).

Step 2: Evaluate the reaction times

After evaluating the above expressions, reaction times would be \( t_1 = 0.05773 \) seconds, \( t_2 = 0.07217 \) seconds, and \( t_3 = 0.06245 \) seconds.

Step 3: Obtain Average Reaction Time

The average reaction time can be calculated by summing the three times obtained and dividing the result by 3, which is the total number of trials. Hence, average time \( t_{avg} = \frac{t_1 + t_2 + t_3}{3} = \frac{0.05773 + 0.07217 + 0.06245}{3}\).

Step 4: Evaluate Average Reaction Time

Finally, the average reaction time would be evaluated as \( t_{avg} = 0.06412 \) seconds.

Key Concepts

Reaction Time Experiment

Understanding the reaction time experiment begins with grasping the basics of human reflexes and response times. In this particular exercise, a simple yet effective method is used to gauge an individual's reaction speed. The setup involves dropping a ruler from a fixed height and recording the distance it falls before being caught.

The key to success in this experiment is consistency in test conditions, such as the starting height of the ruler, the way it's dropped, and how the catcher responds to the drop, which should be solely based on their visual cue. After collecting the data for the distance the ruler falls, the reaction time is calculated using a predetermined formula.

Such experiments are widely used in the field of psychology to study sensorimotor response and can have implications in understanding human performance in various tasks, including driving, sports, and everyday activities requiring quick reflexes.

Quadratic Equations

Quadratic equations form the backbone of many algebraic problems, including the reaction time experiment discussed in this context. A quadratic equation is a second-order polynomial equation in a single variable x with a coefficient a ≠ 0, written in the form of \(ax^2 + bx + c = 0\). The solutions to these equations are known as the roots and can often be found via factoring, completing the square, or applying the quadratic formula.

The given equation in the reaction time experiment, \(t=\sqrt{\frac{d}{192}}\), resembles a quadratic equation as it involves the square root and squares once we manipulate the equation. Understanding quadratic equations is essential for solving complex problems that involve rates, areas, and optimization in various fields such as physics, engineering, and economics.

Average Calculation

Calculation of averages is a fundamental statistical tool used to summarize data. To calculate an average, or mean, we add up all the values and divide by the number of values. In the context of the reaction time experiment, the average calculation is critical in assessing the overall performance over multiple trials.

In our example, the average reaction time over the three trials is found by summing the individual reaction times obtained from the distances the ruler fell and dividing by the total number of trials, which is three in this case. This process gives us a single value representing a typical response time in the set of trials, allowing for a simplified understanding of the collected data. Average calculations are not only crucial in experimental data analysis but also in various fields such as finance, economics, and academics to interpret sets of numbers and make informed decisions.