I would love a solution and an explanation for the following question! The size of a fish varies in time according to the law (dV)/(dt)=-V+(1)/(10)S, where V is the volume of the fish and S is its surface area. For a particular species, the volume and surface area are related to the length of the fish L (in metres) according to V=(L^(3))/(10),

Question

I would love a solution and an explanation for the following question! The size of a fish varies in time according to the law

(dV)/(dt)=-V+(1)/(10)S,

where

V

is the volume of the fish and

S

is its surface area. For a particular species, the volume and surface area are related to the length of the fish

L

(in metres) according to

V=(L^(3))/(10), and ,S=L^(2).

(a) Show that

L

satisfies the differential equation

(dL)/(dt)=(1)/(3)(1-L).

(b) Solve this equation as a linear differential equation to find

L(t)

given that

L=0

when

t=0

. (c) What is the maximum size to which such a fish can grow? (d) If

t

is measured in years, how long does it take for a fish to grow to

50cm

in length?

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