Biomass of haddock: In the study by Beverton and Holt42 described , information is also provided on haddock. We have adapted it for this exercise. We measure length in inches, weight in pounds, and age t in years.
a. It was found that haddock grow to a maximum length of about 21 inches. Furthermore, the difference D = D(t) between maximum length and length L = L(t) at age t is an exponential function. If initially a haddock is 4 inches long, find the initial value of D.
b. It was found that a 6-year-old haddock is about 15.8 inches long. Find a formula that gives L in terms of t.
c. The weight W in pounds of a haddock can be calculated from its length L using
Use function composition to find a formula for weight W as a function of age t.
d. After t years, the total number of fish in the cohort is given by the exponential function
The total biomass of the cohort is the number of fish times the weight per fish. Find a formula for the total biomass B of the cohort as a function of the age of the cohort.
e. How old is the cohort when biomass is at a maximum? (Consider ages up to 10 years.)