Answer :
Certainly! Let's go through this step by step.
### Step 1: Create the Population Model
The population of Detroit in a given year can be modeled by an exponential decay function. Given that the population decreases by 2.8% per year, we can use the general form of the exponential decay model:
[tex]\[ P(t) = P_0 \cdot (1 - r)^t \][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex] years after the initial year,
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the annual rate of decrease,
- [tex]\( t \)[/tex] is the number of years since the initial year.
In this problem:
- [tex]\( P_0 = 951700 \)[/tex] (the population in 2000),
- [tex]\( r = 0.028 \)[/tex] (the annual decrease rate of 2.8%).
### Step 2: Find the Population in 2014
We first need to find the population in 2014, which is 14 years after 2000. Plugging in the values, we have:
[tex]\[ t = 2014 - 2000 = 14 \][/tex]
So, the equation becomes:
[tex]\[ P(14) = 951700 \cdot (1 - 0.028)^{14} \][/tex]
Calculating the population in 2014:
[tex]\[ P(14) \approx 951700 \cdot (0.972)^{14} \][/tex]
[tex]\[ P(14) \approx 639481.1956226993 \][/tex]
Therefore, the population of Detroit in 2014 is approximately [tex]\( 639,481 \)[/tex].
### Step 3: Find When the Population Will Reach 500,000
We want to find the year [tex]\( t \)[/tex] when the population [tex]\( P(t) \)[/tex] reaches 500,000. Using the same exponential decay equation, we set [tex]\( P(t) = 500000 \)[/tex]:
[tex]\[ 500000 = 951700 \cdot (1 - 0.028)^t \][/tex]
To solve for [tex]\( t \)[/tex], we first isolate the exponential term:
[tex]\[ \frac{500000}{951700} = (0.972)^t \][/tex]
Simplifying the fraction:
[tex]\[ \approx 0.5253 = (0.972)^t \][/tex]
To solve for [tex]\( t \)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(0.5253) = t \cdot \ln(0.972) \][/tex]
Then divide both sides by [tex]\( \ln(0.972) \)[/tex]:
[tex]\[ t = \frac{\ln(0.5253)}{\ln(0.972)} \][/tex]
Calculating [tex]\( t \)[/tex]:
[tex]\[ t \approx \frac{-0.6431}{-0.0284} \][/tex]
[tex]\[ t \approx 22.65 \][/tex]
Since [tex]\( t \)[/tex] must be an integer, we round up to the next whole number, [tex]\( t = 23 \)[/tex].
Therefore, the year when the population reaches 500,000 is:
[tex]\[ 2000 + 23 = 2023 \][/tex]
### Final Answers:
a) The population of Detroit in 2014 is approximately 639,481.
b) The population will reach 500,000 in the year 2023.
### Step 1: Create the Population Model
The population of Detroit in a given year can be modeled by an exponential decay function. Given that the population decreases by 2.8% per year, we can use the general form of the exponential decay model:
[tex]\[ P(t) = P_0 \cdot (1 - r)^t \][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex] years after the initial year,
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the annual rate of decrease,
- [tex]\( t \)[/tex] is the number of years since the initial year.
In this problem:
- [tex]\( P_0 = 951700 \)[/tex] (the population in 2000),
- [tex]\( r = 0.028 \)[/tex] (the annual decrease rate of 2.8%).
### Step 2: Find the Population in 2014
We first need to find the population in 2014, which is 14 years after 2000. Plugging in the values, we have:
[tex]\[ t = 2014 - 2000 = 14 \][/tex]
So, the equation becomes:
[tex]\[ P(14) = 951700 \cdot (1 - 0.028)^{14} \][/tex]
Calculating the population in 2014:
[tex]\[ P(14) \approx 951700 \cdot (0.972)^{14} \][/tex]
[tex]\[ P(14) \approx 639481.1956226993 \][/tex]
Therefore, the population of Detroit in 2014 is approximately [tex]\( 639,481 \)[/tex].
### Step 3: Find When the Population Will Reach 500,000
We want to find the year [tex]\( t \)[/tex] when the population [tex]\( P(t) \)[/tex] reaches 500,000. Using the same exponential decay equation, we set [tex]\( P(t) = 500000 \)[/tex]:
[tex]\[ 500000 = 951700 \cdot (1 - 0.028)^t \][/tex]
To solve for [tex]\( t \)[/tex], we first isolate the exponential term:
[tex]\[ \frac{500000}{951700} = (0.972)^t \][/tex]
Simplifying the fraction:
[tex]\[ \approx 0.5253 = (0.972)^t \][/tex]
To solve for [tex]\( t \)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(0.5253) = t \cdot \ln(0.972) \][/tex]
Then divide both sides by [tex]\( \ln(0.972) \)[/tex]:
[tex]\[ t = \frac{\ln(0.5253)}{\ln(0.972)} \][/tex]
Calculating [tex]\( t \)[/tex]:
[tex]\[ t \approx \frac{-0.6431}{-0.0284} \][/tex]
[tex]\[ t \approx 22.65 \][/tex]
Since [tex]\( t \)[/tex] must be an integer, we round up to the next whole number, [tex]\( t = 23 \)[/tex].
Therefore, the year when the population reaches 500,000 is:
[tex]\[ 2000 + 23 = 2023 \][/tex]
### Final Answers:
a) The population of Detroit in 2014 is approximately 639,481.
b) The population will reach 500,000 in the year 2023.
