the population of a town is decreasing at a rate of 1.8% per year. The current population of the own in 12000.
a)write and equation that models the population of the town
b)how long will it take for the population to decline to one quarter of its current population?
c)calculate the approx. instantaneuos rate of change at 5 years.
nope i don't, the school makes it manditory to take advanced functions before calculus
in that case I would take something like
(P(5.01) - P(5))/(5.01 - 5)
= -199
so at the 5 year point, the population is decreasing at appr 200 people/year
a) To write an equation that models the population of the town, you 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 growth rate (expressed as a decimal)
t is the time in years
In this case, the population is decreasing, so the growth rate will be negative. Since the population is decreasing at a rate of 1.8% per year, the growth rate (r) will be -0.018 (expressed as a decimal). The initial population (P₀) is 12000.
b) To find how long it will take for the population to decline to one quarter of its current population, we can set up the equation:
P(t) = P₀ * (1 - r)^t
We want to find the time (t) when P(t) becomes one quarter of P₀. So, our equation becomes:
12000 * (1 - 0.018)^t = 12000 / 4
Simplifying the equation, we have:
0.982^t = 1/4
To solve for t, we can take the logarithm of both sides:
log(0.982^t) = log(1/4)
t * log(0.982) = log(1/4)
t = log(1/4) / log(0.982)
Using a calculator, you can find the value of t.
c) To calculate the approximate instantaneous rate of change at 5 years, we can use calculus. The instantaneous rate of change (also known as the derivative) can be found by taking the derivative of the population equation:
P'(t) = dP/dt = P₀ * (1 - r)^t * ln(1 - r)
Substituting the values, we have:
P'(5) = 12000 * (1 - 0.018)^5 * ln(1 - 0.018)
Calculate the above expression to find the approximate instantaneous rate of change at 5 years.
a) I would write
P(n) = 12000(.982)^n where n is the number of years from the present
for b)
solve .25 = (.982)^n
I assume you know how to do logs.
c) have we established if you know Calculus?
That will determine how to answer that.