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 = 0.007W^(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 14 cm to 20 cm at a constant rate, how fast was this species' brain growing when the average length was 18 cm? (Round your answer to four significant figures.) ? = g/yr

Explanation / Answer

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 24 cm at a constant rate, how fast was this species' brain growing when the average length was 18 cm? Given B = 0.007 W^(2/3) W = 0.12 L^(2.53) It follows that B = 0.007 [0.12 L^(2.53)]^(2/3) B = 7×10^-3×0.12^(2/3)×L^(2.53×2/3) B = 1.703×10^-3 L^(5.06/3) dB/dt = 1.703×10^-3×(5.06/3)×lnL×L^(5.06/3) dL/dt dB/dt = 2.872×10^-3×lnL×L^(5.06/3) dL/dt Since dL/dt = (24 - 16)/10^7 = 0.8×10^-6 cm/year dB/dt = 2.872×10-3×0.8×10-6×lnL×L^(5.06/3) dB/dt = 2.298×10^-9×lnL×L^(5.06/3) g/yr At L = 18 cm, dB/dt = 2.298×10^-9×ln18×18^(5.06/3) = 0.870×10^-6 g/yr The brain growing rate of that fish species at the epoch when the average length measured 18 cm was 0.870 microgram per year.

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