Chapter 11.1, Problem 46E

Dieting A person’s weight depends both on the daily rate of energy intake, say C calories per day, and on the daily rate of the energy consumption, typically between 15 and 20 calories per pound per day. Using an average value of 17.5 calories per pound per day, a person weighing w pounds uses $17.5w$calories per day. If $C=17.5w$, then weight remains constant, and weight gain or loss occurs according to whether C is greater or less than $17.5w$. Source: The College Mathematics Journal.

To determine how fast a change in weight will occur, the most plausible assumption is that $\frac{dw}{dt}$ is proportional to the net excess (or deficit) $C-17.5w$ in the number of calories per day.

a. Assume C is constant and write a differential equation to express this relationship. Use k to represent the constant of proportionality. What does C being constant imply?

b. The units of $\frac{dw}{dt}$ are pounds per day, and the units of $C-17.5w$ are calories per day. What units must k have?

c. Use the fact that 3500 calories is equivalent to 1 lb to rewrite the differential equation in part a.

d. Solve the differential equation.

e. Let $w_0$ represent the initial weight and use it to express the coefficient of $e^{-0.005t}$ in terms of $w_0$ and C.