Brain weight B as a function of body weight W in fish has been modeled by the po

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 16 cm to 21 cm at a constant rate, how fast was this species' brain growing when the average length was 17 cm? (Round your answer to four significant figures.)

Explanation / Answer

Given that:

dL/dt = (21 - 16)/10^7 = 5*10^7

Now

W = 0.12*L^2.53

dW/dt = 0.12*2.53*L^1.53*(dL/dt)

At L = 17 cm

W = 0.12*17^2.53 = 155.6743

dW/dt = 0.12*2.53*17^1.53*5*10^-7 = 1.1584*10^-5

Now

B = 0.007*W^(2/3)

dB/dt = 0.007*(2/3)*W^(-1/3)*(dW/dt)

Using the above values

dB/dt = 0.007*0.667*155.6743^(-1/3)*1.1584*10^-5

dB/dt = 1.0054*10^-8 grams/yr = 10.054 nanograms/year

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