The population of an aquatic species in a certain body of water is approximated

Question

The population of an aquatic species in a certain body of water is approximated by the logistic function P(t)=37,500/1+13e^-0.69t, where t is measured in years. Complete parts a through c. (a) Determine when the population reaches 15,000. It will take 3 years for the population reach 15,000. (Round to the nearest whole number.) (b) Determine when the population is first growing at the rate of 2,500 organisms per year. The population grows at the rate of 2,500 species per year after 1 year(s). (Round to the nearest whole number.) (e) Determine the fastest rate of growth of the species and determine when this occurs. The fastest rate of growth of the species is organisms/year and this occurs after 5 years. (Round to the nearest whole number.)

Explanation / Answer

C)

fastest rate of growth occurs when p'(t) is maximum

from the graph p'(t) is maximum at t =4

p(t)==37500/(1+ 13e^(-0.69t))

p'(t)=(336375. e^(0.69 t))/(e^(0.69 t)+13)^2 ,t=4

fastest rate of growth is 6408 organisms/year

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