Question

A logistic model is given by the equation y= A/1+e−B(x−C). To the nearest hundredth, for what value of x does y equal 0.5?

A) Analyze
B) Solve
C) Estimate
D) Predict

Answer 1

To find the value of x when y equals 0.5 in the given logistic model, we can set y equal to 0.5 and solve the equation for x. The equation x = C - (1/B)ln(2A - 1) represents the value of x when y equals 0.5. Round the result to the nearest hundredth.

Step-by-step explanation:

In the given logistic model, y = A/(1+e^(-B(x-C))). We are asked to find the value of x when y equals 0.5. To solve for x, we can set y equal to 0.5 and solve the equation for x. Here's the step-by-step process:

1. Substitute y = 0.5 in the equation: 0.5 = A/(1+e^(-B(x-C))).
2. Multiply both sides of the equation by (1+e^(-B(x-C))): 0.5(1+e^(-B(x-C))) = A.
3. Divide both sides of the equation by 0.5: 1+e^(-B(x-C)) = 2A.
4. Subtract 1 from both sides of the equation: e^(-B(x-C)) = 2A - 1.
5. Take the natural logarithm of both sides of the equation to eliminate the exponential function: -B(x-C) = ln(2A - 1).
6. Divide both sides of the equation by -B: x-C = -(1/B)ln(2A - 1).
7. Add C to both sides of the equation to isolate x: x = C - (1/B)ln(2A - 1).

Therefore, the value of x when y equals 0.5 is given by the equation x = C - (1/B)ln(2A - 1). Round the result to the nearest hundredth for the final answer.