Question

Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 14 cm to 23 cm at a constant rate, how fast was this species' brain growing when the average length was 19 cm? (Round your answer to four significant figures.)

Answer 1

`(\delta B)/(\delta t) = 1.952\ x\ 10^(-8)\ g/y`

Explanation:

We need to know how fast the brain of the species grows at the point where its average length was 19 cm.

In other words we need to find:

`(\delta B)/(\delta t)`

`B = 0.007W^{(2)/(3)}`

`(\delta B)/(\delta t) = 0.007({(2)/(3)})W^{-(1)/(3)}((\delta W)/(\delta t))`

Now we need to find
`(\delta W)/(\delta t)`

In the statement of the problem it is said that
`(\delta L)/(\delta t)` is constant.

It is also said that the length changed from 14 to 23 cm in
`10 ^ 7` years.

So:

`(\delta L)/(\delta t) = (23-14)/(10^7)`

`(\delta L)/(\delta t) = (9)/(10^7)`

Now we find
`(\delta W)/(\delta t)`

`(\delta W)/(\delta t) = 0.12(2.53)L^(1.53)((\delta L)/(\delta t))\\\\(\delta W)/(\delta t) = 0.12(2.53)L^(1.53)((9)/(10^7))\\`

Now we find W and
`(\delta W)/(\delta t)` for L = 19

`W = 0.12(19)^(2.53)\\\\W = 206.27`

`(\delta W)/(\delta t) = 0.12(2.53)(19)^(1.53)((9)/(10^7))\\\\(\delta W)/(\delta t) = 2.4719\ x\ 10^(-5)`

Now replace
`(\delta W)/(\delta t)` and W in the main equation of
`(\delta B)/(\delta t)`

`(\delta B)/(\delta t) = 0.007({(2)/(3)})(206.27)^{-(1)/(3)}(2.4719\ x\ 10^(-5))\\\\(\delta B)/(\delta t) = 1.952\ x\ 10^(-8)\ g/y`