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Problem 51 Easy Difficulty

Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance) If $ L_1(t) $ and $ L_2(t) $ are the sizes of two organs in an organism of age $ t, $ then $ L_1 $ and $ L_2 $ satisfy an allometric law if there specific growth rates are proportional:
$ \frac {1}{L_1} \frac {dL_1}{dt} = k \frac {1}{L_2} \frac {dL_2}{dt} $
where $ k $ is a constant.
(a) Use the allometric law to write a differential equation relating $ L_1 $ and $ L_2 $ and solve it to express $ L_1 $ as a function of $ L_2. $
(b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating $ B $ (cell biomass) and $ V $ (cell volume) was found to be $ k = 0.0794. $ Write $ B $ as a function of $ V. $

Answer

a) $$L_{1}=K L_{2}^{k}, \text { where } K=e^{C}$$
b) $B(V)=c V^{0.0794}$

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Video Transcript

All right. So we were told that we have two quantities eggs and wine in there, related by the Cella metric formula. Why is equal to in X to the down and then we wantto figure out the how they're raised of change are late. But we want to know how do I do t and d x d t. And this is just a classic related race for only a relationship between two quantities. We want to relate their rates of change. So we just differentiate with respect, Tio. So we're told. Actually first too. Take the natural longer them of both sides to get the relationship we want. So if we take the natural ogle over them, we get natural. Why equals the natural log of and plus in times natural log of X? And here I am just using properties and natural longer than so long about product a lot longer in them of a product. It's just that some of the logger thems and then if I have log of of something to power the power he come down. Okay, so now you can differentiate. So the derivative of log of something is one over that something times the derivative of that something just United team natural law. Giving in is a constants, that zero. And then here we have him, time's derivative of natural law. Get something against just one over that something. Times three X City and we have relationship that we want.