The population, P, of a certain species of yeast cells grows according to the mo

Question

The population, P, of a certain species of yeast cells grows according to the model P(t) = -t^2 + 12t + 20. In this formula, the population is measured in thousands of bacteria and the time in hours. Find the average growth rate between t = 1 and t = 3 hours Find the formula for the instantaneous growth rate of the population using the formula for the average growth rate and then setting delta t to zero. Find the time when the instantaneous growth rate of the population is zero. Find the initial instantaneous growth rate of the population. Bonus. Find the time when the population becomes extinct (disappears) according to this model.

Explanation / Answer

a) average growth rate =[P(3)-P(1)]/(3-1)

=[(-32+12*3 +20)-(-12+12*1 +20)]/2

=[(-9+36+20)-(-1+12+20)]/2

=16/2

=8

average growth rate = 8 thousand /hour

b) average growth rate =[P(t+t)-P(t)]/t

average growth rate =[(-(t+t)2+12(t+t) +20)-(-t2+12t +20)]/t

average growth rate =[-t2-(t)2-2tt+12t+12t +20+t2-12t -20]/t

average growth rate =[-(t)2-2tt+12t ]/t

average growth rate =-t-2t+12

instantaneous growth rate = limt->0-t-2t+12

instantaneous growth rate = -0-2t+12

instantaneous growth rate = -2t+12

c) instantenous growth rate =0

-2t +12=0

t =6 hours

d)at t =0,initial instantanous growth rate =-2*0+12 =12 thousand/hour

e)population extinct when -t2+12t +20=0

t2-12t -20=0

t=[12+(122-4*1*(-20))]/2

t=[12+224)]/2

t=13.483

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