Question

The population of a city in the Northeast is given by
p(t) = 130(1 + 12e−0.02t)−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
g(p) = 2p − 0.001p3 garbage trucks
where p is the population of the city in thousands. Evaluate each expression in 2013. (Round your answers to three decimal places.)

Answer 1

Populations can be represented by a function.

The population of the city in 2013 is 1598.130 thousand, and the number of garbage truck needed is 4078458.918.

The functions are represented as:

$\mathbf{P(t) = 130(1 + 12e^{-0.02t}) - 1}}$

$\mathbf{g(p) = 2p - 0.001p^3}$

In 2013, t = 3.

So, we have:

$\mathbf{P(3) = 130(1 + 12e^{-0.02 \times 3}) - 1}}$

$\mathbf{P(3) = 130(1 + 12e^{-0.06}) - 1}}$

$\mathbf{P(3) = 130(1 + 11.301) - 1}}$

$\mathbf{P(3) = 130(12.301) - 1}}$

$\mathbf{P(3) = 1598.130}$

So, the number of garbage trucks in 2013 is:

$\mathbf{g(1598.130) = 2 \times 1598.130- 0.001 \times 1598.130^3}$

$\mathbf{g(1598.130) = -4078458.918}$

Hence, the population of the city in 2013 is 1598.130 thousand, and the number of garbage truck needed is 4078458.918.

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