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Chapter 3: Problem 67

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?

Short Answer

Expert verified
The weight one can lift will follow a logistic function type, typically expressed as \( f(x) = \frac{C}{1+ae^{-bx}} \). In this case, the data points can be collected each week to create an exact logistic function. This function may then be used to project future weights that could be lifted with time \(x\). Adjustments need to be made over time to increase the accuracy of the model's predictions.

Step by step solution

01

Model Type

Recognize that the function description indicates a logistic growth function. The progress is quick at first, then starts to slow down, which is characteristic of a logistic function. A logistic function is typically expressed as \( f(x) = \frac{C}{1+ae^{-bx}} \) where \(x\) is the time (in weeks), \(f(x)\) is the weight you can lift, \(C\) is the maximum weight you will ever be able to lift, \(a\) is a constant that depends on your initial strength, and \(b\) is a constant that determines how fast you get stronger.
02

Collecting Data

Over several weeks, track the maximum weight you can lift at the end of each week. This will provide the necessary data points to characterize your logistic growth function. You then have pairs of values for the time \(x\) (in weeks) and the corresponding \(f(x)\).
03

Function Estimation

Use the data points you collected to estimate the logistic function that best fits your data. There are many ways to do this, but one common way is to use a least-squares fit. This will provide the constants \(a\), \(b\), and \(C\) for your logistic function.
04

Predict Future Performances

Now, you can use your function to predict the future. Plug in the number of weeks you want to predict for \(x\) and compute \(f(x)\) to get the predicted maximum weight you will be able to lift. Remember, this is just a model and actual results may vary.
05

Validation and Adjustment

Over time, compare the predictions of your model with your actual performance, and adjust the constants \(a\), \(b\), and \(C\) as necessary to improve the fit of the model.

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