Problem 63 - College Algebra Essentials

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

The situation described fits a logarithmic growth model. A logarithmic function of the form \( f(x) = a + b\log(c(x-h)) \) needs to be obtained by fitting the data of the maximum weight lifted each week. This function can then be used to predict future performance by substitifying the week number into the function.

Step 1: Understand the type of growth

Given the nature of the problem where there is a rapid growth initially which then slows down and almost levels off, it is clear that this is a case of logarithmic growth. So, the function that we need to obtain has to be a logarithmic function, which can be written in the general form as \( f(x) = a + b\log(c(x-h)) \).

Step 2: Gather data

Collect the maximum number of pounds lifted each week. The weeks will be our input (x) and the weight lifted will be the output (f(x)). Logarithmic growth often appears when the rate of change slows down over time, so it is expected in this case.

Step 3: Fit the data to the function

Fit the data to the function using a suitable method like least squares method. Use the obtained model to determine the parameters a, b, c and h for the function \( f(x) = a + b\log(c(x-h)) \). The software or calculator used will provide these parameters, which characterizes the logarithmic function.

Step 4: Predict future values

Use the obtained function to make future predictions by substituting the week number into the function. This will provide an estimate of the maximum weight that can be lifted in that week.