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**MathBane** The question:

For this exercise, round all regression parameters to three decimal places.

In the fishery sciences it is important to determine the length of a fish as a function of its age. One common approach, the von Bertalanffy model, uses a decreasing exponential function of age to describe the growth in length yet to be attained; in other words, the difference between the maximum length and the current length is supposed to decay exponentially with age. The following table shows the length *L*, in inches, at age *t*, in years, of the North Sea sole.

t = age L = Length

1 |3.7

2 |7.5

3 |10

4 |11.5

5 |12.7

6 |13.5

7 |14

8 |14.4

Suppose the maximum length attained by the sole is 15.0 inches.

(a) Make a table showing, for each age, the difference *D* between the maximum length and the actual length *L* of the sole.

t = age D = Difference

1 |11.3

2 | 7.5

3 | 5

4 | 3.5

5 | 2.3

6 | 1.5

7 | 1

8 |0.6

(I was only able to get row 8 by using the information underneath the previous table.) Why?

(b) Find the exponential function that approximates *D*. (Round all regression parameters to three decimal places.)

*D* = 17.466(.662)^t

(c) Find a formula expressing the length *L* of a sole as a function of its age *t*. (Round all parameters to three decimal places.)

*L* = a + b ln (t)

L = 3.939 + 5.261 ln (t)

(I thought this would be $\displaystyle 4.877(1.176)^t$, but that was incorrect.)