Brain weight bb as a function of body weight ww in fish has been modeled by the power function b=.007w2/3b=.007w2/3, where bb and ww are measured in grams. a model for body weight as a function of body length ll (measured in cmcm) is w=.12l2.53w=.12l2.53. if, over 1010 million years, the average length of a certain species of fish evolved from 15cm15cm to 20cm20cm at a constant rate, how fast was the species' brain growing when the average length was 18cm18cm? round your answer to the nearest hundredth.

You are to find the rate of change of B with respect to t, dB/dt. Differentiate the first equation with respect to t.

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

We have to determine W and dW/dt.

For W when L = 18 cm, we use the second equation

W = 0.12 (18^2.53) = 1,800 g

For dW/dt, differentiate the second equation with respect to t.

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

We still have to find dL/dt from the given information. The solution would be:

dL/dt = (20 cm - 15 cm) / 10,000,000 years = 5*10⁻⁷ cm/y

So,

dW/dt = 0.12 (2.53) (18^1.53) (5*10⁻⁷) = 1.26*10⁻⁵ g/y

Now that we know W and dW/dt, we can finally solve for dB/dt.

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

dB/dt = 0.007 (2/3) (1800^-1/3) (1.26*10⁻⁵) = 1.04*10⁻⁸ g/y