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__

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 * (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 * 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.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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. $\newline$ $P(t)=\square$

View step-by-step help

Home

Math Problems

Algebra 2

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.