Problem 102 - Elementary and Intermediate Algebra

Question

After participating in an eight-week weight training program, a college student was selected to be part of a study to see how much strength he would lose if he discontinued working out. The function \(M_{B}(w)=225-14 \ln (4 w+1)\) approximates his maximum bench press (in pounds) \(w\) weeks after stopping weight training. What was his maximum bench press: a. At the end of the eight-week training course? b. 6 weeks after stopping the weight training?

Step-by-Step Solution

Answer

a. 225 lbs; b. approximately 179.94 lbs

Step 1: Understand the function

The function given is \(M_{B}(w)=225-14 \ln (4w+1)\), which approximates the student's maximum bench press in pounds, \(w\) weeks after stopping weight training.

Step 2: Determine maximum bench press at the end of the course

At the end of the eight-week training course, the weight training has not yet stopped. Therefore, we calculate \(M_{B}(0)\) to find his maximum bench press's initial state.Substitute \(w = 0\) into the function:\[ M_{B}(0) = 225 - 14 \ln (4\times0 + 1) \]Simplifying gives:\[ M_{B}(0) = 225 - 14 \ln (1) \]Since \(\ln(1) = 0\), it results in:\[ M_{B}(0) = 225 \]

Step 3: Determine bench press 6 weeks after stopping

Substitute \(w = 6\) into the function to find the maximum bench press after 6 weeks:\[ M_{B}(6) = 225 - 14 \ln (4\times6 + 1) \]Calculate inside the logarithm:\[ 4\times6 + 1 = 25 \]Thus, the equation becomes:\[ M_{B}(6) = 225 - 14 \ln (25) \]Using a calculator, approximate \(\ln(25)\) which is around 3.2189:\[ M_{B}(6) = 225 - 14 \times 3.2189 \]Calculate the final result:\[ M_{B}(6) \approx 225 - 45.0646 \approx 179.9354 \]

Key Concepts

Mathematical Modeling

Mathematical modeling helps us represent real-world scenarios using mathematical concepts and structures. It involves creating equations that depict relationships in an understandable way. In this exercise, our model predicts changes in the student's bench press ability post-training.
The given function is a model that relies on the logarithmic function to simulate how strength diminishes when training ceases. The equation used is:

• \( M_{B}(w) = 225 - 14\ln(4w + 1) \)

The first term, 225, represents the initial maximum bench press, the value before the student stops training.
A logarithmic function, like the one here, is effective in modeling scenarios where changes happen gradually, such as strength loss or radioactive decay. This model captures the gradual decline due to muscle disuse effectively, illustrating how mathematical modeling can transform complex scenarios into simpler forms that are easy to analyze and predict.

Exponential Decay

Exponential decay processes describe scenarios where quantities decrease at a rate proportional to their current value. In simple terms, things decrease faster when they are larger and slow down as they get smaller.
In the context of this exercise, the logarithmic component contributes to the decay effect. While the function itself, \( 225 - 14 \ln(4w + 1) \), is not a pure exponential decay, it behaves similarly with respect to how quickly the bench press maximum decreases post-training.
This slower initial drop in strength mirrors natural processes like cooling and natural depreciation, where the logarithmic function's properties help capture the non-linear decline effectively. Understanding this can help link mathematical concepts to physical intuitions in real-world situations.

Function Evaluation

Function evaluation refers to plugging numbers into a function to get a relevant output. It's about determining the exact value the function expresses under certain conditions.
To evaluate the student's maximum bench press after the eight-week course, we substitute \( w = 0 \) into our model:

• \( M_{B}(0) = 225 - 14 \ln(1) = 225 \)

At this time, strength remains maximal since no decay has occurred yet.
Six weeks post-training, we substitute \( w = 6 \) into the function:

• Calculate \( M_{B}(6) = 225 - 14\ln(25) \)

Using a calculator for \( \ln(25) \), we find it to be approximately 3.2189, leading to:

• \( M_{B}(6) \approx 179.9354 \)

This precise evaluation shows how the student's bench press decreases in value, illustrating the strength's decay over six weeks.