Problem 118 - Algebra and Trigonometry

Question

The function \(P(x)=95-30 \log _{2} x\) models the percentage, \(P(x),\) of students who could recall the important features of a classroom lecture as a function of time, where \(x\) represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the graph of the function. Use this information to solve Exercises \(117-118\). Round answers to one decimal place. After how many days have all students forgotten the important features of the classroom lecture? (Let \(P(x)=0\) and solve for \(x\).) Locate the point on the graph that conveys this information.

Step-by-Step Solution

Answer

After approximately 8.9 days, all students have forgotten the important features of the classroom lecture. The corresponding point on the graph would be (8.9, 0).

Step 1: Write Down the Equations

We know that \(P(x) = 95 - 30 \log_{2}(x)\), and we are looking for the times \(x\) at which \(P(x) = 0\). Let's set the equation \(P(x) = 0\).

Step 2: Solve the Equation

Solving the equation \(95 - 30 \log_{2}(x) = 0\), we get \(30 \log_{2}(x) = 95\). Now we divide both sides by 30 to isolate the logarithm: \(\log_{2}(x) = 95 / 30 = 3.16.\)

Step 3: Determine the Value of X

To find \(x\), we need to find the antilogarithm of \(\log_{2}(x) = 3.16\), or raise 2 to the power of 3.16, \(x = 2^{3.16}\).

Step 4: Calculate the Value of X

We can now calculate \(x = 2^{3.16}\), after rounding it to one decimal place, we get \(x = 8.9\) days.

Step 5: Locate the Point on the Graph

To locate this point on the graph, we simply need to find the point where \(x = 8.9\). The \((x, y)\) representation of the point on the graph would be (\(8.9, 0\)), as \(P(x) = 0\) indicates that all students have forgotten the important features of the lecture.

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that help us understand how a quantity grows or decays at an exponential rate. In the context of our problem, we use the logarithmic function to model how the percentage of information students retain from a lecture changes over time.
This particular function is a base-2 logarithm, denoted as \( \log_{2}(x) \). It's used to find the exponent to which the base (2 in this case) must be raised to get the value \( x \).

• In our equation \( P(x) = 95 - 30 \log_{2}(x) \), 95 represents the percentage of retention at the starting point, right after the lecture.
• The term \(-30 \log_{2}(x)\) describes how the recall percentage decreases over time, using a logarithmic decay model.

The concept of solving the exponential equation by logarithms involves finding the number of days, \( x \), where this term will decrease the recall to 0%, which indicates that the knowledge has been forgotten. By setting \( P(x) = 0 \) and solving for \( x \), we're able to determine how quickly information is lost based on its initial retention rate and the decay model.

Percentage Recall

Percentage recall is a measure of how much information is remembered or retained over time. In educational settings, understanding how memory retention works is crucial for designing effective teaching methods.
In our given function \( P(x) = 95 - 30 \log_{2}(x) \), percentage recall is modeled to show how the proportion of students remembering a lecture decreases logarithmically over time.

• The initial percentage of students recalling the lecture is 95%, implying high retention right after learning.
• However, this number decreases following a logarithmic scale, which reflects the nature of forgetting, where memory retention tends to drop steeply at first and then levels off.

By setting the function to zero and solving for \( x \), we gain insight into when the recall diminishes entirely. This provides valuable information about the natural decline in memory retention, emphasizing the need to revisit or reinforce learning material regularly to maintain high recall rates.

Student Learning Retention

Student learning retention is the process and ability to retain knowledge after initially learning it. Retention rates can vary widely across different subjects and individuals. The mathematical model presented in this problem provides a generic insight into how learning retention changes over time, helping us understand its dynamics better.
Description of the model we use:

• It predicts when the retention of information approaches zero, which is when students entirely forget the material taught.
• In the given equation, fluctuations in retention are mapped using a logarithmic function, showing that learning retention follows an exponential decay pattern.

This kind of understanding is essential for educators because it highlights the importance of reinforcement strategies, such as additional review sessions and practice tests, to bolster student retention. Tailoring educational practices to counteract natural forgetting curves can lead to more effective learning outcomes and longer-term retention of knowledge.