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A population of fish was added to a lake. The population size [tex]P(t)[/tex] of this species can be modeled by the following function, where [tex]t[/tex] is the number of years since the species was added to the lake.

[tex]\[ P(t) = \frac{1500}{1 + 3 e^{-0.34 t}} \][/tex]

Find the population size of the species after 4 years and after 7 years. Round your answers to the nearest whole number as necessary.

Population size after 4 years: [tex]\square[/tex] fish

Population size after 7 years: [tex]\square[/tex] fish



Answer :

GinnyAnswer
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]

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