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10 October, 16:11

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.

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  1. 10 October, 16:16
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    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
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