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]
