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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If L1(t) and L2(t) are the sizes of two organs in an organism of age t, then L1 and L2 satisfy an allometric law if their specific growth rates are proportional:

1 L 1 d L 1 d t = k 1 L 2 d L 2 d t

where k is a constant.

(a) Use the allometric law to write a differential equation relating L1 and L2 and solve it to express L1 as a function of L2.

(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.

(a)

To determine

To solve: The solve the given function: 1L1dL1dt=k1L2dL2dt .

Explanation

Given:

L1(t) and L2(t) be the size of two organs of age t.

The allometric function is, 1L1dL1dt=k1L2dL2dt

Calculation:

The function can be written as follows,

ddt(lnL1)=ddt(lnL2k)

Integrate on both sides,

ddt(lnL1)=ddt(lnL2k

(b)

To determine

To find: The function B(V) .

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