Population Growth A major corporation is building a...

Population Growth A major corporation is building a 4,325-acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Cove's population (in thousands) t years from now will be given by P(t) = (30t^2 + 60t + 120) / (t^2 + 2t + 8) (a) Find the rate (in people/year) at which Glen Cove's population is changing with respect to time. P'(t) = (b) What will the population be after 10 years? (Round your answer to the nearest person.) people At what rate (in people/year) will the population be increasing when t = 10? (Round your answer to the nearest integer.) people/year

Transcript

00:01 Hello, let's have a look on the question.

00:03 So a function for population is given to us 30 t squared plus 60t plus 120 divided by t squared plus 2t plus 8.

00:16 So this is the function of population.

00:19 We need to find out p -t -t.

00:21 So this will be equal to denominator as it is multiplied by differentiation of numerator, that is, 60 t plus 60 plus numerator as it is that is 30 t square minus numerator as it is that is 30 t square plus 60 t plus 120 into differentiation of the denominator that is 2 t plus 2 divided by square of denominator so t square plus 2 t plus 8 whole square now this is equal to 60 tq plus 120 t square plus 60 into 60 x2x2x2 t squared minus 60 into 8 is 480 t plus 60 t squared plus 120 t plus 480 minus 2 into 30 that is 60 this is tq plus 2 into 60 that is 120 t squared plus 2 into 120 that is 240 t plus 2 into 30 t squared is 60 t squared plus 120 t plus 120 now divided by denominator whole square that is t square plus 2 t plus 8 whole square we can write 60 t q plus 120 and 60 t square this will be 180 t square plus 480 t and 120 480 and 120 t will be equal to 600 t plus we have 480 minus 60 t cube minus 180 t square minus 380 t and minus 380 t and minus 240 divided by t squared plus 2 t plus 8 whole square now this values will be cancelled out and the remaining values will be equal to 220 t minus this is plus 240 divided by t squared plus 2 t plus 8 whole square now for the b part we need to find out the population after 10 years so p t was equal to 30 t squared plus 60 t plus 120 divided by t squared plus 2t plus 8.

03:33 Now in place of t we can put 10.

03:36 So population after 10 years will be equal to 30 into 100 plus 60 into 10 plus 120 divided by 100 plus 200 plus 200 plus 8.

03:49 So this will be equal to 3 ,000 plus 600 plus 120 divided by 308.

04:00 So this population after 10 years will be equal to 12 .077.

04:07 Now we need to find out the rate at which population is changing after 10 years, then p -t -t will be equal to...

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