Weight Loss Courtney tried several different diets over a...

Weight Loss Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight $W$ followed a fluctuating curve that could be mod- eled by the function $W=220-1.5 t+9.81 \sin (\pi t / 4)$ , where $t$ was measured in months after January 1 of the first year and $W$ was measured in pounds. (a) What was Courtney's weight at the start and at the end of two years? (b) What was her maximum weight during the two-year period? (e) What was her minimum weight during the two-year period?

Weight Loss Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight $W$ followed a fluctuating curve that could be mod- eled by the function $W=220-1.5 t+9.81 \sin (\pi t / 4)$ ,
where $t$ was measured in months after January 1 of the first year and $W$ was measured in pounds.
(a) What was Courtney's weight at the start and at the end of
two years?
(b) What was her maximum weight during the two-year period?
(e) What was her minimum weight during the two-year period?

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Key Concepts

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions, such as functions, to represent real-world phenomena. This involves crafting equations that incorporate variable factors, periodic behavior, and trends to simulate how a system changes over time.

Extreme Value Analysis

Extreme value analysis involves finding the maximum and minimum values of a function within a given domain. This is achieved either by evaluating the function at boundary points or by using calculus techniques to determine critical points where the function’s rate of change is zero.

Sinusoidal Functions

Sinusoidal functions, which are based on sine and cosine, are used to model periodic or oscillating phenomena. They allow the representation of fluctuations around a central value, explaining how variables such as weight can exhibit regular, cyclical variations over time.

Function Evaluation

Evaluating functions at specific points is a basic concept in mathematics that involves calculating the output for given input values. In practical applications, this allows one to determine the value of a variable at critical moments by substituting these time values into the function.

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