Chapter 8.4, Problem 23ES

Fish is a rich source of protein and omega-3 fatty acids. However, some fish, such as shark, swordfish, and tuna, contain high concentrations of methylmercury and should be eaten only in moderation. For example, suppose a man starts a new diet in which he consumes 425 grams of albacore tuna per week (roughly equivalent to three 5 oz cans). His body will absorb about 145 micrograms $\left(\mu g\right)$ of methylmercury from the tuna. On the hand, approximately $9\%$ of any given quantity of methylmercury within his system will be eliminated in a week. Assume that there are 300 $\mu g$ of methylmercury in his body when he begins the diet and that afterwards his tuna consumption is the only source of additional amounts of the compound. Let $y(t)$ denote the number of micrograms of methylmercury in his body $t$ weeks after the start of his diet.

(a) Find an initial-value problem whose solution is $y(t).$

(b) Find a formula for y(t).

(c) Suppose the man weighs 160 lb. According to U.S EPA guidelines, the amount of methylmercury within his body should be at or below $\text{569}\mu g$ to avoid toxicity. Based upon your solution to part (b), determine whether the man should modify his diet.