The length of some fish is modeled by a von Bertalanffy growth function. For Pac

Question

The length of some fish is modeled by a von Bertalanffy growth function. For Pacific halibut, this function has the form L(t) = 200(1 - 0.956e^-0.18t) where L(t) is the length (in centimeters) of a fish t years old. (a) What is the length of a new-born halibut at birth? cm (b) Use the formula to estimate the length of an 8-year-old halibut. () cm (c) At what age would you expect the halibut to be 110 cm long? () years old (d) What is the practical (physical) significance of the number 200 in the formula for L(t)? This represents the average length of Pacific halibut. This represents the average length of adolescent Pacific halibut. This represents the length of a new-born Pacific halibut. This represents the maximum possible length of Pacific halibut. This represents the difference in length between male and female Pacific halibut.

Explanation / Answer

L(t) = 200 ( 1 - 0.956 e^-0.18t)

a) length of a new born halibut is

plug t = 0

200 ( 1 - 0.956 e^0 )

8.80 cm

b) length of 8 year old halibut

plug t = 8

200 ( 1 - 0.956 e^-0.18(8))

= 154.70 cm

c) at what age length is 110 cm

110 = 200 ( 1 - 0.956 e^-0.18t)

dividing both sides by 200

.55 = ( 1 - 0.956 e^-0.18t)

subtracting 1 from both sides

-.45 = - 0.956 e^-0.18t

dividinh both sides by -0.956

.4707 = e^-0.18t

ln .4707 = -0.18 t

t = ln .4707/ -0.18

t = 4.19 years old approx

d) 200 is the maximum possible length of pacific halibut

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