Practice Problems for 2/27
Based on information collected about Body Mass Index, one study found that the percentage of the US population which is considered obese (according to their BMI) is given by
\[
P(t)=\frac{4}{10^4}t^3+\frac{36}{10^4}t^2+\frac{8}{10}t+12,\quad 0 \leq t \leq 13.
\]
(So the study covers a 13 year period.)
The study advocates a drastic changes to laws concerning school cafeterias and PE programs, as well implementing things like a sugar tax, stating,
"Not only is the population of obese individuals increasing, but the rate at which the US population is becoming obese is also rising."
Based on the information provided by the study is this statement true or false?
They claim that the
rate of the
rate of change of the percentage is rising. So, in calculus this means they make the claim \(P''(t)>0\).
\[
P'(t)=\frac{12}{10^4}t^2+\frac{72}{10^4}t+\frac{8}{10},\]
and
\[
P''(t)=\frac{d}{dt}P'(t)=\frac{24}{10^4}t+\frac{72}{10^4}.
\]
Since we can only plug in \(t\in [0,13]\), \(P''(t)>0\), and according to this model, the claim is true.
Note that one should always analyze models to make sure they accurately reflect reality. While this study may have some good information and their claim is true, one could argue that a strict cut off for BMI to determine obesity is not the best metric, so this model doesn't tell the whole story.