A lake formed by a newly constructed dam is stocked with 1,000 fish. Their population is expected to increase according to the logistic curve

N=30/(1+29e^-1.35t)
where N is the number of fish, in thousands, expected after t years. The lake will be open to fishing when the number of fish reaches 20,000. How many years, to the nearest year, will this take?

well, just solve for t:

20 = 30/(1+29e^-1.35t)
1+29e^-1.35t = 1.5
e^-1.35t = 0.5/29
-1.35t = -4.060
t = 3

To find out how many years it will take for the number of fish to reach 20,000, we need to solve the equation:

30 / (1 + 29e^(-1.35t)) = 20

Let's solve this equation step by step:

1. Multiply both sides of the equation by (1 + 29e^(-1.35t)):
30 = 20(1 + 29e^(-1.35t))

2. Divide both sides of the equation by 20:
30/20 = 1 + 29e^(-1.35t)

3. Simplify the left side of the equation:
1.5 = 1 + 29e^(-1.35t)

4. Subtract 1 from both sides of the equation:
1.5 - 1 = 29e^(-1.35t)

5. Simplify the left side of the equation:
0.5 = 29e^(-1.35t)

6. Divide both sides of the equation by 29:
0.5/29 = e^(-1.35t)

7. Take the natural logarithm (ln) of both sides of the equation:
ln(0.5/29) = -1.35t

8. Divide both sides of the equation by -1.35:
t = ln(0.5/29) / -1.35

Now, let's calculate the value of t using a calculator:

t ≈ -0.5 years (rounded to the nearest year)

Since a negative value for t doesn't make sense in this context, it means that the number of fish will never reach 20,000 according to the given logistic curve equation.

To find out how many years it will take for the number of fish to reach 20,000, we can set up the equation as follows:

20 = 30 / (1 + 29e^(-1.35t))

To solve for t, we can start by isolating the exponential term by multiplying both sides of the equation by (1 + 29e^(-1.35t)):

20(1 + 29e^(-1.35t)) = 30

Now, we can simplify the equation:

20 + 580e^(-1.35t) = 30

Next, let's isolate the exponential term:

580e^(-1.35t) = 10

Now divide both sides of the equation by 580:

e^(-1.35t) = 10/580

Simplify the right side of the equation:

e^(-1.35t) = 1/58

To solve for t, we'll take the natural logarithm (ln) of both sides:

ln(e^(-1.35t)) = ln(1/58)

The natural logarithm and exponential function cancel each other out:

-1.35t = ln(1/58)

Now divide both sides of the equation by -1.35:

t = ln(1/58) / -1.35

Using a calculator, we can compute the value:

t ≈ 8.32 years

Therefore, it will take approximately 8 years (to the nearest year) for the number of fish to reach 20,000.