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The lion population in a certain reserve drops by
5% every year. Currently, the population's size is 200.
Write a function that gives the lion population size,
P(t),
t years from today.

P(t)=◻++x__

The lion population in a certain reserve drops by $5 \%$ every year. Currently, the population's size is $200$ . $\newline$ Write a function that gives the lion population size, $P(t)$ , $t$ years from today. $\newline$ $P(t)=\square$

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Interpret the exponential function word problem

Full solution

Q. The lion population in a certain reserve drops by

5 \%

every year. Currently, the population's size is

200

.

\newline

Write a function that gives the lion population size,

P(t)

,

t

years from today.

\newline

P(t)=\square

Exponential Decay Function: Step $1$ : To model a population that decreases by a certain percentage each year, we use an exponential decay function. The general form of an exponential decay function is $P(t) = P_0 \times (1 - r)^t$ , where $P_0$ is the initial population size, $r$ is the decay rate as a decimal, and $t$ is the time in years.
Initial Population and Decay Rate: Step $2$ : The initial population size, $P_0$ , is given as $200$ lions. The annual decay rate is $5\%$ , which as a decimal is $0.05$ . Therefore, the decay factor is $1 - r$ , which is $1 - 0.05 = 0.95$ .
Substitute Values into Function: Step $3$ : Substitute the values of $P_0$ and the decay factor into the exponential decay function to get the function for the lion population size. $P(t) = 200 \times 0.95^t$ .
Check Population Decrease: Step $4$ : Check the function to ensure it represents a decreasing population over time. If $t$ increases, the term $0.95^t$ decreases because $0.95$ is less than $1$ , which means the population size $P(t)$ will also decrease. This aligns with the given information that the population drops by $5\%$ each year.

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