Nutrients in low concentrations inhibit growth of an organism, but high concentr

Question

Nutrients in low concentrations inhibit growth of an organism, but high concentrations are often toxic. Let c be the concentration of a particular nutrient (in moles/liter) and P be the population density of an organism (in number/cm2). Suppose that it is found that the effect of this nutrient causes the population to grow according to the equation:

P(c)=(1300c)/(1+625c^2)

Find the concentration of the nutrient that yields the largest population density of this organism and what the population density of this organism is at this optimal concentration.

Optimal nutrient concentration=

Largest population density=

Explanation / Answer

P(c) = 1300c/(1 +625c2)

At maximum density P '(c) = 0

==> [1300(1 +625c2) - 1300c(1250c)]/(1 +625c2)2 = 0                           since (u/v)' = [u'v -uv']/v2

==> [1300 + 812500c2 - 1625000c2] = 0

==> 1300 = 812500c2

==> c = (1300/812500)

==> c = 1/25 = 0.04

Hence optimal nutrient concentration = 0.04 moles/liter

P(0.04) = 1300(0.04)/(1 +625(0.04)2)

==> P(0.04) = 26

Hence largest population density = 26 number/cm2

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