Problem 42 - College Algebra

Question

You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?

Step-by-Step Solution

Answer

To describe the maximum lifting weight over time, a logistic function can be used, which typically models scenarios with an initial rapid growth that eventually levels off. After collecting lifting data and adjusting the function parameters, predictions on future lifting weights can be made by inputting the week number into the function. However, it must not be overlooked that this model assumes a continuous, unmodified training pattern and does not consider potential physical condition variations.

Step 1: Understand Logistic Growth

Logistic growth is a common growth form which is represented by the logistic function. A logistic function is graphically characterized by initial exponential growth which eventually levels off and becomes nearly constant.

Step 2: Formulate the Logistic Function

The general form of a logistic function is \(f(x) = \frac{c}{1 + a \cdot e^{-b \cdot x}}\), where \(c\) is the carrying capacity or maximum number, in this case the weight measurable, \(a\) is related to the initial amount, \(b\) represents the grow rate, and \(e\) is the base of the natural logarithm exponential function.

Step 3: Collect and Plug-in Data

Collected data such as maximum lift per each week needs to be inserted into the function. Since this function typically requires tools like software for best fit parameter determination, assumingly, necessary adjustments to the logistic function can be made using appropriate coefficients.

Step 4: Use Function for Prediction

Once the function is properly adapted to the collected data points, it will be used to predict future outcomes. As long as the lifting training continues with the same patterns, by inputting the week number into the function it can estimate the maximum weight expected to be lifted on that week. But keep in mind that this function won't be accurate indefinitely, as it assumes conditions remain the same, and it doesn't account for potential plateaus or peaks in physical condition.