Answer: a) To find the exponential decay rate, we can use the formula:
A = A0 * e^(kt)
where A is the final population, A0 is the initial population, e is the base of the natural logarithm, k is the decay rate, and t is the time in years.
Given that the population declined from 382,000 in 2010 to 303,000 in 2017, we have:
A = 303,000
A0 = 382,000
t = 2017 - 2010 = 7 years
Substituting these values into the formula, we get:
303,000 = 382,000 * e^(7k)
Simplifying the equation, we divide both sides by 382,000:
0.7932 = e^(7k)
To solve for k, we take the natural logarithm (ln) of both sides:
ln(0.7932) = ln(e^(7k))
Using the property of logarithms, we can bring down the exponent:
ln(0.7932) = 7k * ln(e)
Since ln(e) is equal to 1, the equation becomes:
ln(0.7932) = 7k
Now we can solve for k by dividing both sides by 7:
k = ln(0.7932) / 7
Approximately, k ≈ -0.0543
Therefore, the exponential decay rate is approximately -0.0543.
To write the function that represents the population of the town t years after 2010, we use the formula:
P(t) = A0 * e^(kt)
where P(t) is the population at time t.
In this case, the function becomes:
P(t) = 382,000 * e^(-0.0543t)
b) To estimate the population of the town in 2021, we need to find t, the number of years after 2010. Since 2021 is 11 years after 2010, we substitute t = 11 into the function:
P(11) = 382,000 * e^(-0.0543 * 11)
Calculating this, we find that the estimated population of the town in 2021 is approximately 276,304.
c) To find the year when the population will reach 293,000, we need to solve for t in the function:
293,000 = 382,000 * e^(-0.0543t)
Dividing both sides by 382,000, we get:
0.7665 = e^(-0.0543t)
Taking the natural logarithm of both sides, we have:
ln(0.7665) = ln(e^(-0.0543t))
Using the property of logarithms, we bring down the exponent:
ln(0.7665) = -0.0543t * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(0.7665) = -0.0543t
Now we solve for t by dividing both sides by -0.0543:
t = ln(0.7665) / -0.0543
Approximately, t ≈ 11.64
Therefore, the population will reach 293,000 approximately 11.64 years after 2010.
Step-by-step explanation: Hope this helps broskie :D