Sure, let's solve this problem step by step using the given mathematical function.
The population size [tex]\( P(t) \)[/tex] of the species is defined by the function:
[tex]\[ P(t) = \frac{1500}{1 + 3e^{-0.34t}} \][/tex]
To find the population size after 4 years and 7 years, we simply need to substitute [tex]\( t = 4 \)[/tex] and [tex]\( t = 7 \)[/tex] into this function and evaluate the results.
### Step 1: Calculate the population size after 4 years
Substitute [tex]\( t = 4 \)[/tex] into the function:
[tex]\[ P(4) = \frac{1500}{1 + 3e^{-0.34 \cdot 4}} \][/tex]
First, calculate the exponent part:
[tex]\[ -0.34 \cdot 4 = -1.36 \][/tex]
Now, find [tex]\( e^{-1.36} \)[/tex].
After calculating the value of [tex]\( e^{-1.36} \)[/tex], you will have a number that you'll plug back into the function:
[tex]\[ P(4) = \frac{1500}{1 + 3 \cdot e^{-1.36}} \][/tex]
Now, simplify the denominator:
[tex]\[ P(4) = \frac{1500}{1 + (3 \cdot \text{some value from } e^{-1.36})} \][/tex]
Finally, evaluate this expression to find the population size after 4 years. Rounding the result to the nearest whole number, we get:
[tex]\[ P(4) \approx 847 \][/tex]
### Step 2: Calculate the population size after 7 years
Substitute [tex]\( t = 7 \)[/tex] into the function:
[tex]\[ P(7) = \frac{1500}{1 + 3e^{-0.34 \cdot 7}} \][/tex]
First, calculate the exponent part:
[tex]\[ -0.34 \cdot 7 = -2.38 \][/tex]
Now, find [tex]\( e^{-2.38} \)[/tex].
After calculating the value of [tex]\( e^{-2.38} \)[/tex], you will have a number that you'll plug back into the function:
[tex]\[ P(7) = \frac{1500}{1 + 3 \cdot e^{-2.38}} \][/tex]
Now, simplify the denominator:
[tex]\[ P(7) = \frac{1500}{1 + (3 \cdot \text{some value from } e^{-2.38})} \][/tex]
Finally, evaluate this expression to find the population size after 7 years. Rounding the result to the nearest whole number, we get:
[tex]\[ P(7) \approx 1174 \][/tex]
### Final Population Sizes
The population size after 4 years is approximately:
[tex]\[ \boxed{847} \, \text{fish} \][/tex]
The population size after 7 years is approximately:
[tex]\[ \boxed{1174} \, \text{fish} \][/tex]
