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Select the correct answer.

Identify the exponential function that correctly fits the situation below.

The size of a rainforest is currently decreasing at a rate of [tex]50\%[/tex] per year. If there are currently 210,000 square miles of rainforest, then about how many square miles of rainforest will there be in [tex]t[/tex] years?

A. [tex]F = 210,000(0.5)^t[/tex]

B. [tex]F = 220,000(1.4)^t[/tex]

C. [tex]F = 210,000(0.95)^t[/tex]

D. [tex]F = 210,000(1.5)^t[/tex]

Answer :

To model a decrease of a quantity by 50% per year, we start with an initial area of the rainforest, which is given as

[tex]$$
A_0 = 210\,000.
$$[/tex]

Since the area is decreasing by 50% each year, this means that every year the area is multiplied by 0.5. Thus, after 1 year the area becomes

[tex]$$
210\,000 \times 0.5 = 105\,000.
$$[/tex]

Following this pattern, after [tex]$t$[/tex] years the area can be described by the exponential function

[tex]$$
F(t) = 210\,000 \cdot (0.5)^t.
$$[/tex]

After comparing with the given multiple-choice options, we see that the correct function is

[tex]$$
\boxed{F = 210\,000(0.5)^t.}
$$[/tex]

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