The body mass index (BMI) measures body weight in relation...

The body mass index (BMI) measures body weight in relation to height. A BMI of 25 to 29.9 is considered overweight, a BMI of 30 or more is considered obese, and a BMI of 40 or more is morbidly obese. The percentage of the U.S. population that is obese is approximated by the function $$ P(t)=0.0004 t^{3}+0.0036 t^{2}+0.8 t+12 \quad(0 \leq t \leq 20) $$ where $t$ is measured in years, with $t=0$ corresponding to the beginning of $1991 .$ a. What percentage of the U.S. population was deemed obese at the beginning of $1991 ?$ At the beginning of $2010 ?$ b. How fast was the percentage of the U.S. population that is deemed obese changing at the beginning of 1991? At the beginning of 2010? (Note: A formula for calculating the BMI of a person is given in Exercise 35.) Source: Centers for Disease Control and Prevention.

The body mass index (BMI) measures body weight in relation to height.
A BMI of 25 to 29.9 is considered overweight, a BMI of 30 or more is considered obese, and a BMI of
40 or more is morbidly obese. The percentage of the U.S. population that is obese is approximated by the function
$$
P(t)=0.0004 t^{3}+0.0036 t^{2}+0.8 t+12 \quad(0 \leq t \leq 20)
$$
where $t$ is measured in years, with $t=0$ corresponding to the beginning of $1991 .$
a. What percentage of the U.S. population was deemed obese at the beginning of $1991 ?$ At the beginning of $2010 ?$
b. How fast was the percentage of the U.S. population that is deemed obese changing at the beginning of 1991? At the beginning of 2010?
(Note: A formula for calculating the BMI of a person is given in Exercise 35.)
Source: Centers for Disease Control and Prevention.

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Key Concepts

Function Evaluation

Evaluating a function at specific values involves substituting particular inputs into the function to obtain corresponding outputs. This process is crucial for determining the model’s predictions at given points in time, such as the beginning of certain years.

Differentiation and Instantaneous Rate of Change

Differentiation is a fundamental concept in calculus that calculates the instantaneous rate of change of a function. By finding the derivative of a model, one can assess how quickly the modeled quantity—in this case, the obesity percentage—is changing at any specific moment in time.

Body Mass Index (BMI) and Obesity Classification

BMI is a numerical value derived from an individual's weight and height and is used as a screening tool to categorize weight status (e.g., normal, overweight, obese). This concept is important in public health for assessing and categorizing the prevalence of obesity in a population.

Mathematical Modeling with Functions

Mathematical modeling involves using functions to represent real-world phenomena. Here, a polynomial function is used to model the percentage of a population that is obese over time, allowing predictions and analysis of trends based on a given time variable.

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