f(x)=110/1+8e-2x'
The logistic growth model is often used to model population growth, where the population grows rapidly at first, but eventually levels off as it approaches its carrying capacity.
In the given function, f(x) represents the population size at time x. The function is in the form of the logistic growth model, where the carrying capacity is 110 and the growth rate is 8e-2.
To find the population size at a specific time, substitute the value of x into the function. For example, to find the population size at time x=10, we have:
f(10) = 110 / (1 + 8e-2(10))
= 110 / (1 + 8e-1)
= 110 / (1 + 0.8)
= 60.44
Therefore, the population size at time x=10 is approximately 60.44.
To find the value of f(x) in the given equation f(x) = 110/(1 + 8e^(-2x)), you can simply substitute the value of x into the equation and evaluate it.
Let's say you want to find the value of f(x) when x = 5. Here's how you can do it:
1. Substitute the value of x into the equation:
f(5) = 110 / (1 + 8e^(-2*5))
2. Simplify the expression inside the parentheses:
f(5) = 110 / (1 + 8e^(-10))
3. Calculate the value of e^-10:
e^-10 is approximately 4.5399929762484854 × 10^(-5).
4. Substitute the value of e^-10 into the equation:
f(5) ≈ 110 / (1 + 8(4.5399929762484854 × 10^(-5)))
5. Calculate the expression inside the parentheses:
f(5) ≈ 110 / (1 + 3.6319943809987883 × 10^(-4))
6. Add 1 to the value inside the parentheses:
f(5) ≈ 110 / (1.0003631994380999)
7. Evaluate the division:
f(5) ≈ 109.96401704664997
Therefore, when x = 5, f(x) ≈ 109.96401704664997.
To understand the steps, let's break down the given function:
f(x) = 110 / (1 + 8e^(-2x))
Step 1: Start with the function f(x) = 110 / (1 + 8e^(-2x)).
Step 2: Notice that the function involves exponential notation. e is the base of the natural logarithm.
Step 3: Simplify the expression within the parentheses by evaluating the exponent: e^(-2x).
Step 4: Rewrite the expression as f(x) = 110 / (1 + 8 * e^(-2x)).
Step 5: Evaluate e^(-2x). This is the exponential function with the base e raised to the power of -2x.
Step 6: Simplify the expression by substituting the value of e^(-2x) into the function.
Step 7: Rewrite the expression as f(x) = 110 / (1 + 8 * (1 / e^(2x))).
Step 8: Evaluate 1 / e^(2x). This is equivalent to e^(-2x).
Step 9: Substitute the value of 1 / e^(2x) into the function.
Step 10: Rewrite the expression as f(x) = 110 / (1 + 8 / e^(2x)).
Step 11: Simplify further by multiplying both the numerator and denominator by e^(2x).
Step 12: Rewrite the expression as f(x) = (110 * e^(2x)) / (1 * e^(2x) + 8).
Step 13: Distribute the multiplication to get f(x) = (110 * e^(2x)) / (e^(2x) + 8).
Step 14: Simplify the expression further if needed based on the given context or any specific question.