Respuesta :
Answer:
[tex]21[/tex] nanogram/year
Explanation:
The given equation for body weight as a function of length
[tex]W= 0.12*L^{2.53}[/tex]
Differentiating both sides of the equation with respect to length , we get -
[tex]\frac{dW}{dt} = 0.12*2.53*l^{1.53}*\frac{dl}{dt} \\\frac{dW}{dt} = 0.3036*l^{1.53}*\frac{dl}{dt}\\[/tex]------Equation (A)
As we know -
[tex]\frac{dl}{dt} = \frac{5}{10^7}[/tex]
substituting the value of [tex]\frac{dl}{dt}[/tex] in equation (A), we get -
[tex]\frac{dW}{dt} = 0.3036*17^{1.53}*5*10^{-7}\\= 1.158*10^{-5}[/tex]----Equation (C)
On differentiating both side of under given equation, we get -
[tex]B = 0.007*W^{\frac{2}{3}}\\\frac{dB}{dt} = 0.007*\frac{2}{3} *W^{\frac{-1}{3}}*\frac{dW}{dt}[/tex]-----Equation (D),
Substituting the value of W wrt t in Equation (D), we get -
[tex]\frac{dB}{dt} = 0.007*\frac{2}{3} *W^{\frac{-1}{3}}*\frac{dW}{dt}\\\frac{dB}{dt} = 0.007*\frac{2}{3}*17^{\frac{-1}{3}}* 1.158*10^{-5}\\\frac{dB}{dt} = 2.10*10^{-8}[/tex]
[tex]= 21[/tex] nanogram/year
The brain grows at a rate of 0.0041 grams per million of years when the body length is 17cm.
How to see how fast the brain was growing?
We know that the brain's weight is given by:
[tex]B = 0.007*W^{2/3}[/tex]
And the weight, given as a function of L, the body length, is given by:
[tex]W = 0.12*L^2[/tex]
By replacing that in the brain weight equation we get:
[tex]B = 0.007*(0.12*L^2)^{2/3} = 0.0017*L^{4/3}[/tex]
In 10 million years, the length evolved from 16cm to 23cm, then the rate of change of L is:
(23cm - 16cm)/10 = 0.7 cm per million of years.
Then the brain grows at a rate of:
[tex]B' = (4/3)*0.0017*L^{1/3}*L'[/tex]
And we know L' = 0.7 cm per million of years. Ignoring the units (because we don't have the units for the equations, we can see that when L = 17cm the brain weight increases at:
[tex]B' = (4/3)*0.0017*(17)^{1/3}*0.7 = 0.0041[/tex]
This would be in grams per million of years.
If you want to learn more about rates, you can read:
https://brainly.com/question/8728504
