The population of a city in the Northeast is given by p ( t ) = 130(1 + 12 e 0.0
ID: 2859788 • Letter: T
Question
The population of a city in the Northeast is given by
p(t) = 130(1 + 12e0.02t)1 thousand people
where t is the number of years since 2010. The number of garbage trucks needed by the city can be modeled as
______
where p is the population of the city in thousands. Evaluate each expression in 2014. (Round your answers to three decimal places.)
(a)
p(t)
10.764 thousand people
(b)
g(p)
20.281 garbage trucks
(c)
_____ thousand people per year
(d)
_____ trucks per thousand people
(e)
_____ garbage trucks per year
Explanation / Answer
The population of a city in the Northeast is given by
p(t) = 130(1 + 12e0.02t)1 thousand people
We need to find population in 2014
ie, t= 4
p(4) = 130(1 + 12e0.02(4))1 = 10.764thousands
(c).
d/dt (130(1 + 12e0.02t)1 )
= (-156 e^(t/20) ) / 5(12e^(t/50 +1)^2
dp/dt at t = 4
= - 0.1725 thousand (decreasing)
(b) g(p) = 2p 0.001p^3 garbage trucks
g(10.764) = 2(10.764) 0.001(10.764)^3 garbage trucks
= 21.528 - 1.247
= 20.281 garbage trucks
(d) dg/dp = d/dp( 2p 0.001p^3)
= 2 - 0.003 p^2
dg/dp at p = 10.764
= 2 - 0.003 (10.764)^2
= 2- 0.3476
= 1.652
dg/dt = d/dt( 2p 0.001p^3)
= 2 dp/dt - 0.003p^2 dp/dt
= 2( - 0.1725) - 0.003(10.764)^2 (-0.1725)
= - 0.285
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