Related rates problem

Question

Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W^2⁄3,

 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 = 0.12L^2.53.

 If, over 10 million years, the average length of a certain species of fish evolved from 13 cm to 23 cm at a constant rate, how fast (in g/yr) was this species' brain growing when its average length was 19 cm? (Round your answer to four significant figures.)

Answer 1

Hi Emmy,

Your question is asking for the rate of change of the brain weight. Based on the variables defined in your problem, this is "dB/dt".

The question also tells you that the length of the fish grew from 13cm to 23cm (an increase of 10cm) over the course of 10,000,000 years. This means that the rate of change of the length is 10/10,000,000, or 1/1,000,000. In terms of your problem, this is dL/dt.

We have an equation for B in terms of W. We also have W in terms of L. Since we want dB/dt and we have dL/dt, we want to generate an equation for B in terms of L.

B = 0.007*W^2/3

W = 0.12*L^2.53

Substituting, we get

B = 0.007*(0.12*L^2.53)^2/3

B = 0.007*(0.2433*L^1.69)

B = 0.0017031*L^1.69

To generate dB/dt and dL/dt, we can implicitly differentiate this equation with respect to t.

d/dt(B = 0.0017031*L^1.69)

dB/dt = 0.00287824*L^0.69*dL/dt

The question is interested in the brain's growth rate when the length of the fish was 19cm. So, L = 19.

dB/dt = 0.00287824*(19)^0.69*1/1,000,000

dB/dt = 2.195 x 10^-8 grams/ year

Hope this helps!