How to solve this problem step by step with application of integration?

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1 Answer
Alvin L.
Mar 4, 2018

#a)# #200#
#b)# After #(3+sqrt129)/2~~7.18# years
#c)# #100# individuals #-># extinct after #5# years
#color(white)(c.)400# individuals #-># extinct after #8# years

Explanation:

We begin by solving for #P(t)#. To do this, we can integrate both sides of the following relation using the power rule:

#P'(t)=30-20t#

#int\ P'(t)\ dt=int\ 30-20t\ dt#

#P(t)=30t-10t^2+C#

To solve for #C#, we can use the fact that #P(0)=300# (the initial population):

#P(0)=30*0-10*0+C=300#

#C=300#

We see that the constant will just be the initial population.

This gives that our function #P(t)# is:

#P(t)=-10t^2+30t+300#

We can find the population after five years by plugging in #t=5#:

#P(5)=-10*5^2+30*5+300=200#

Next we want to find out when the population goes extinct, which is when #P(t)=0#. We can solve for when this happens using the quadratic formula.

#-10t^2+30t+300=0#

#(-10t^2)/(-10)+(30t)/(-10)+300/(-10)=0/(-10)#

#t^2-3t-30=0#

#t=(3+-sqrt(129))/2#

Since we are only looking for positive solutions (we can't go back in time), we can discard the negative solution.

#t=(3+sqrt129)/2~~7.18#

We recognized before that the constant would just be whatever the initial population was, so we can work out the other extinction times just by changing the constant in the equation:

#-10t^2+30t+100=0->t=5#

#-10t^2+30t+400=0->t=8#

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