Sure! Let's solve this problem step-by-step using the provided model [tex]\( P(t) = 448,000 e^{-0.014t} \)[/tex], where [tex]\( t \)[/tex] is the number of years from the present.
### Step 1: Identify the given parameters
- Initial population [tex]\( P(0) \)[/tex]: 448,000
- Decay rate: [tex]\(-0.014\)[/tex]
- Time [tex]\( t \)[/tex]: 7 years
### Step 2: Setup the population model for 7 years
We use the formula [tex]\( P(t) = 448,000 e^{-0.014t} \)[/tex] to find the population in 7 years. Substituting [tex]\( t = 7 \)[/tex] into the formula:
[tex]\[ P(7) = 448,000 e^{-0.014 \times 7} \][/tex]
### Step 3: Calculate the exponent
The exponent part of the equation is calculated as follows:
[tex]\[ -0.014 \times 7 = -0.098 \][/tex]
### Step 4: Evaluate the exponential part
Now we evaluate the exponential function [tex]\( e^{-0.098} \)[/tex].
### Step 5: Find [tex]\( P(7) \)[/tex]
We multiply the initial population by the evaluated exponential part:
[tex]\[ P(7) = 448,000 \times e^{-0.098} \][/tex]
### Step 6: Calculate the numerical result
After evaluating the expression, we get:
[tex]\[ P(7) \approx 406,178.7088817566 \][/tex]
### Step 7: Rounding to the nearest person
Rounding [tex]\( 406,178.7088817566 \)[/tex] to the nearest person gives us:
[tex]\[ P(7) \approx 406,179 \][/tex]
So, the predicted population in 7 years, according to the given model, is approximately 406,179 people.
