Question

application exercises
with aging, body fat increases and muscle mass declines. the line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. use the graphs to solve exercises 99 - 106.

source:thompson et al., the science of nutrition, benjamin cummings, 2008

1. state the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
2. state the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
3. for what age does the percent body fat in women reach a maximum? what is the percent body fat for that age?
4. at what age does the percent body fat in men reach a maximum? what is the percent body fat for that age?
5. use interval notation to give the domain and the range for the graph of the function for women.
6. use interval notation to give the domain and the range for the graph of the function for men.
7. the function p(x)= - 0.002x² + 0.15x + 22.86 models percent body fat, p(x), where x is the number of years a persons age exceeds 25. use the graphs to determine whether this model describes percent body fat in women or in men.

Explanation:

Step1: Analyze women's body - fat graph

For women, observe the slope of the graph. The graph is increasing on the interval $(25,45)$ and decreasing on the intervals $(45,75)$.

Step2: Analyze men's body - fat graph

For men, the graph is increasing on the interval $(25,55)$ and decreasing on the interval $(55,75)$.

Step3: Find women's maximum

For women, the maximum percent body - fat occurs at $x = 45$ years of age. The maximum percent body - fat is $36$.

Step4: Find men's maximum

For men, the maximum percent body - fat occurs at $x = 55$ years of age. The maximum percent body - fat is $24$.

Step5: Find women's domain and range

The domain of the function for women (age) is $[25,75]$. The range (percent body - fat) is $[28,36]$.

Step6: Find men's domain and range

The domain of the function for men (age) is $[25,75]$. The range (percent body - fat) is $[20,24]$.

Step7: Determine the model

Substitute some values of age (related to $x$) from the graphs into the model $p(x)=-0.002x^{2}+0.15x + 22.86$. By comparing with the trends and values in the graphs, we find that this model describes percent body - fat in men.

Answer:

1. The percent body fat in women is increasing on the interval $(25,45)$ and decreasing on the interval $(45,75)$.
2. The percent body fat in men is increasing on the interval $(25,55)$ and decreasing on the interval $(55,75)$.
3. The percent body fat in women reaches a maximum at age $45$ and the maximum percent body fat is $36$.
4. The percent body fat in men reaches a maximum at age $55$ and the maximum percent body fat is $24$.
5. Domain: $[25,75]$, Range: $[28,36]$
6. Domain: $[25,75]$, Range: $[20,24]$
7. The model $p(x)=-0.002x^{2}+0.15x + 22.86$ describes percent body fat in men.