Question

2.

Suppose the population of kangaroos in a different province of Australia is modeled by
the following, with t = 0 representing the year 2020:
P() = 76(0.92)
a.) Using a calculator, find P(10) and interpret it in the context of the problem. Be
specific.
b.) What is happening to the population of kangaroos in this problem? What specific value
in the equation causes this? By what percentage is the kangaroo population decreasing
each year?
c.) As the value of t increases to very large numbers, do you think this model will be
realistic? Explain. Using a calculator, find P(100) to help support your answer.

Answer 1

a) 30 kangaroos in 2030

b) decreasing 8% per year

c) large t results in fractional kangaroos: P(100) ≈ 1/55 kangaroo

Explanation:

We assume your equation is supposed to be ...

P(t) = 76(0.92^t)

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a) P(10) = 76(0.92^10) = 76(0.4344) = 30.01 ≈ 30

In the year 2030, the population of kangaroos in the province is modeled to be 30.

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b) The population is decreasing. The base 0.92 of the exponent t is the cause. The population is changing by 0.92 -1 = -0.08 = -8% each year.

The population is decreasing by 8% each year.

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c) The model loses its value once the population drops below 1/2 kangaroo. For large values of t, it predicts only fractional kangaroos, hence is not realistic.

P(100) = 75(0.92^100) = 76(0.0002392)

P(100) ≈ 0.0182, about 1/55th of a kangaroo