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An animal population is modeled by the function A(t) that satisfies the differential equation dA dt equals the product of A divided by 1125 and the quantity 450 minus A . What is the animal population when the population is increasing most rapidly?
45 animals 225 animals 40 animals 180 animals

Sagot :

ajeigbeibraheem

Answer:

225 animals

Step-by-step explanation:

From the given information:

[tex]\dfrac{dA}{dt} = ( \dfrac{A}{1125}) (450 - A)[/tex]

[tex]\dfrac{dA}{dt} = \dfrac{450A - A^2}{1125}[/tex]

For a population increasing most rapidly; we have:

[tex](\dfrac{dA}{dt} \implies max = A')[/tex]

Thus; for [tex]\dfrac{d}{dA} ( \dfrac{dA}{dt}) \implies \dfrac{450-2A}{1125}[/tex]

[tex]\dfrac{dA'}{dA} \implies 0 \\ \\ \\ A = 225[/tex]

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