Chapter 2.3, Problem 8P

Cholesterol is produced by the body for the construction of cell walls, and is also absorbed from certain foods. The blood cholesterol level is measured in units of milligrams per decilitre, or $\text{mg/dl}$. The net cholesterol production or destruction by the body is modelled by a rate $r$ per day times the difference between the body’s “natural” cholesterol level(a constant) and the actual cholesterol level at any time $t$. The rate of absorption from food is estimated as a constant $k$ in milligrams per decilitre per day.

A person’s cholesterol level at the start of the testing period is $150\text{mg/dl}$. Find an expression for the cholesterol level at any subsequent time $t$. If the rate is $0.10$ per day and the natural level is $100\text{mg/dl}$, find the cholesterol level of the person $10$ days after the start of the testing period in terms of $k$.

If $k=25$, what is the cholesterol level of this person after a long time?

Suppose this person starts a low-cholesterol diet. What must the value of $k$ be so that the long-time cholesterol level is $180\text{mg/dl}$.