Problem 146 - College Algebra Essentials

Question

This group exercise involves exploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) \(7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

Step-by-Step Solution

Answer

The analysis will require function evaluation, graph plotting and then analyzing these graphs which will help in understanding the differences and similarities between the height growths of boys and girls.

Step 1: Function Evaluation

Let's start with the function for boys' height growth, \(f(x)=29+48.8 \log (x+1)\), for \(x=5,6,7, \ldots, 15\). Substitute each of these values into the function and calculate the corresponding function values. Do the same for the girls’ model function \(g(x)=62+35 \log (x-4)\) making sure to ignore x values that create undefined function (logarithm of negative number or zero).

Step 2: Graph Plotting

Once all the function values have been calculated from Step 1, plot these points on a graph for both functions. Be careful to denote x-axis with years and y-axis with the percentage of adult height attained. Connect these plotted points with a smooth curve.

Step 3: Graph Analysis

After the graphs are ready, inspect them and compare the two. Focus on their growth pace, where they intersect if at all and their shape to understand the development differences.