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

Question:

Brain weight {eq}B{/eq} as a function of body weight {eq}W{/eq} in a species of fish has been modeled by the power function {eq}B = 0.007W^{\frac{2}{3}}{/eq}, where {eq}B{/eq} and {eq}W{/eq} are measured in grams. Also, a model for body weight as a function of body length L (measured in centimeters) is {eq}W = 0.08L^{2.52}{/eq}.

a) Give an equation expressing brain weight {eq}B{/eq} in terms of body length {eq}L{/eq}.

b) Assuming that all of these quantities are functions of time, differentiate your answer to part (a) with respect to time, giving an equation that relates the rates of change of brain weight and body length.

c) If, over 10 million years, this species of fish increased from 15 cm to 20 cm in length, at a constant rate, find the rate of change of their brain weight now.

Rate of change of a function

The rate of change of a function determines how fast the value of a function is changing with time. Rate of change can vary at different points of time, Hence, the rate of change is also a function of time usually and is determined by taking the first derivative of the function with respect to time.

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According to the information in the question, for a fish of weight W and length L

Brain Weight Model {eq}B = 0.007W^{\frac{2}{3}} {/eq},

Body... Calculating & Interpreting a Function's Average Rate of Change

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Chapter 9 / Lesson 8
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The average rate of change in a given function is helpful to identify speeds and other changing variables. Learn why these are used, and how they are calculated and interpreted through examples.