AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Exponential Decay Population Prediction for a Town
Mathematics
Grade 9 (Junior High School)
Question Content
A town has a population of 24,000. The population is expected to decrease 0.5% annually for the next 20 years. Write a function that represents this situation. Then predict the population in 20 years. Round your answer to the nearest whole person.
Correct Answer
Function: $P(t) = 24000(0.995)^t$; Population in 20 years: 21694
Detailed Solution Steps
1
Step 1: Write the exponential decay function. The general form is $P(t) = P_0(1-r)^t$, where $P_0$ is the initial population, $r$ is the annual decay rate, and $t$ is time in years.
2
Step 2: Substitute the given values: $P_0=24000$, $r=0.5\\%=0.005$. So the function is $P(t) = 24000(1-0.005)^t = 24000(0.995)^t$.
3
Step 3: Calculate the population after 20 years by substituting $t=20$: $P(20) = 24000(0.995)^{20}$.
4
Step 4: Compute $(0.995)^{20}≈0.903917$, then multiply by 24000: $24000×0.903917≈21694$.
Knowledge Points Involved
1
Exponential Decay for Population Modeling
Used to model decreasing quantities like population with constant annual decay. The formula is $P(t)=P_0(1-r)^t$, where $P_0$ is initial population, $r$ is decimal decay rate, and $t$ is time.
2
Converting Percentages to Decimals
To use a percentage rate in an exponential function, divide the percentage by 100 to get its decimal equivalent. For 0.5%, this is $0.5/100=0.005$.
3
Evaluating Exponential Expressions
To find the value of an exponential function at a specific time, substitute the time value into the function and calculate using a calculator, then round to the appropriate precision.
Loading solution...
