The growth of fish has been shown to satisfy a model given by the von Bertalanffy equation:

L(t) = L0(1 - e-bt),

where L0 and b are constants that fit the data. In Math 121, it was shown that there is often an allometric model relating the weight and length of different animals. A model relating the weight of a fish as a function of its length

W(L) = kLa,

where k and a are constants that fit the data.

a. Below are growth data for the Blue Marlin (Makaira mazara) [1].

Age (yr)
1
2
3
4
5
6
7
8
9
10
Length (cm)
83
155
208
249
279
302
319
332
342
349

Find the least squares best fit of the data to the von Bertalanffy equation above. Give the values of the constants L0 and b (to at least 3 significant figures) and write the model with these constants. Find the intercepts and any asymptotes for the length of the Blue Marlin. Graph the data and the model.

b. Below are data on the length and weight for the Blue Marlin [2].

Length (cm)
110
135
160
165
175
190
210
225
255
300
Weight (kg)
13
41
50
53
62
95
119
173
248
426

Use Excel's Trendline (Power Law) to find an allometric model of the form above. Give the value of the constants k and a (to at least 3 significant figures) and write the model with these constants. Graph the data and the model.

c. Create a composite function to give the weight of the Blue Marlin as a function of its age, W(t). Find the intercepts and any asymptotes for W(t). Graph the weight of a Blue Marlin as it ages.

d. Find the derivative of W(t) using the chain rule. Also, compute the second derivative, then determine when this second derivative is zero. From this information, find at what age the Blue Marlin are increasing their weight the most and determine what that weight gain is. Graph W '(t).

[1] M. G. Hinton. Status of Blue Marlin in the Pacific Ocean. Website accessed 1/04.
[2] J. H. Uchiyama and T. K. Kazama. Updated Weight-on-length relationaships for pelagic fishes caught in the central north Pacific Ocean and bottomfishes from the Northwestern Hawaiian Islands, www.nmfs.hawaii.edu/adminrpts/PIFSC_Admin_Rep_03-01.pdf, (accessed 1/04)