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.)

Accepted Solution

Answer:[tex]\frac{\delta B}{\delta t} = 1.952\ x\ 10^{-8}\ g/y[/tex]Step-by-step 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:[tex]\frac{\delta B}{\delta t}[/tex][tex]B = 0.007W^{\frac{2}{3}}[/tex][tex]\frac{\delta B}{\delta t} = 0.007({\frac{2}{3}})W^{-\frac{1}{3}}(\frac{\delta W}{\delta t})[/tex]Now we need to find [tex]\frac{\delta W}{\delta t}[/tex]In the statement of the problem it is said that [tex]\frac{\delta L}{\delta t}[/tex]  is constant.It is also said that the length changed from 14 to 23 cm in [tex]10 ^ 7[/tex] years.So:[tex]\frac{\delta L}{\delta t} = \frac{23-14}{10^7}[/tex][tex]\frac{\delta L}{\delta t} = \frac{9}{10^7}[/tex]Now we find [tex]\frac{\delta W}{\delta t}[/tex][tex]\frac{\delta W}{\delta t} = 0.12(2.53)L^{1.53}(\frac{\delta L}{\delta t})\\\\\frac{\delta W}{\delta t} = 0.12(2.53)L^{1.53}(\frac{9}{10^7})\\[/tex]Now we find W and [tex]\frac{\delta W}{\delta t}[/tex] for L = 19[tex]W = 0.12(19)^{2.53}\\\\W = 206.27[/tex][tex]\frac{\delta W}{\delta t} = 0.12(2.53)(19)^{1.53}(\frac{9}{10^7})\\\\\frac{\delta W}{\delta t} = 2.4719\ x\ 10^{-5}[/tex]Now replace  [tex]\frac{\delta W}{\delta t}[/tex] and W in the main equation of [tex]\frac{\delta B}{\delta t}[/tex][tex]\frac{\delta B}{\delta t} = 0.007({\frac{2}{3}})(206.27)^{-\frac{1}{3}}(2.4719\ x\ 10^{-5})\\\\\frac{\delta B}{\delta t} = 1.952\ x\ 10^{-8}\ g/y[/tex]