the population of preston is 89000 and is decreasing by 1.8% each year
A:Write a function that models the population as a function of time t
b predict when the population will be 50000
To write a function that models the population as a function of time t, we can use the formula for exponential decay:
P(t) = P₀ * (1 - r)^t
Where:
P(t) is the population at time t,
P₀ is the initial population,
r is the decay rate (as a decimal),
t is the time in years.
In this case, the initial population is 89000 and the decay rate is 1.8% or 0.018 (as a decimal). Therefore, the function becomes:
P(t) = 89000 * (1 - 0.018)^t
To predict when the population will be 50000, we need to solve for t in the equation P(t) = 50000:
50000 = 89000 * (1 - 0.018)^t
We can solve this equation using logarithms. Take the logarithm (base 10 or natural logarithm) of both sides:
log(50000) = log(89000 * (1 - 0.018)^t)
log(50000) = log(89000) + log((1 - 0.018)^t)
Now, divide both sides by log(1 - 0.018):
log(50000) / log(1 - 0.018) = t
By substituting the values and solving this equation, we can find the time t when the population will be 50000.
To model the population as a function of time t, we can use the exponential decay formula:
P(t) = P0 * (1 - r)^t
Where:
- P(t) is the population at time t
- P0 is the initial population at t = 0
- r is the annual rate of decrease as a decimal
- t is the number of years
In this case, the initial population (P0) is 89000, and the rate of decrease (r) is 1.8% or 0.018.
Therefore, the function that models the population would be:
P(t) = 89000 * (1 - 0.018)^t
To predict when the population will be 50000, we can rearrange the equation:
50000 = 89000 * (1 - 0.018)^t
To solve for t, we need to use logarithms. Taking the natural logarithm (ln) of both sides:
ln(50000) = ln(89000 * (1 - 0.018)^t)
Using the logarithmic property, we can simplify the equation:
ln(50000) = ln(89000) + t * ln(1 - 0.018)
Now, we can isolate t by subtracting ln(89000) from both sides:
t * ln(1 - 0.018) = ln(50000) - ln(89000)
Finally, divide both sides by ln(1 - 0.018) to solve for t:
t = (ln(50000) - ln(89000)) / ln(1 - 0.018)
Using a calculator, you can find the value of t which will give you the predicted number of years until the population reaches 50000.
decrease of 1.8% is a remaining portion of 98.2%
So, the population after t years is
89000 * 0.982^t
now just find t when the function = 50000