31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
a. How long will it take to reduce the number of cases to 1000?

Verified step by step guidance

1. Identify the type of decay model used to represent the reduction in the number of disease cases. Since the number of cases decreases by a fixed percentage each year, this is an example of exponential decay.
2. Write the general exponential decay formula: \(N(t) = N_0 \times (1 - r)^t\), where \(N(t)\) is the number of cases after \(t\) years, \(N_0\) is the initial number of cases, and \(r\) is the decay rate (expressed as a decimal).
3. Substitute the given decay rate of 25% (which is \(r = 0.25\)) into the formula, so it becomes \(N(t) = N_0 \times (0.75)^t\).
4. Set \(N(t)\) equal to 1000 (the target number of cases) and solve for \(t\): \(1000 = N_0 \times (0.75)^t\). This equation will allow you to find the time needed to reduce the cases to 1000.
5. Take the natural logarithm of both sides to solve for \(t\): \(\ln(1000) = \ln(N_0) + t \times \ln(0.75)\). Then isolate \(t\) by rearranging the equation to \(t = \frac{\ln(1000) - \ln(N_0)}{\ln(0.75)}\).

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In this problem, the number of disease cases reduces by a fixed percentage each year, which models exponential decay. Understanding this helps in setting up the decay formula to predict future values.

Decay Rate and Decay Factor

The decay rate is the percentage by which the quantity decreases per time period, here 25%. The decay factor is 1 minus the decay rate (0.75), representing the fraction of cases remaining after each year. This factor is used in the exponential decay formula to calculate the number of cases over time.

Solving for Time in Exponential Decay

To find how long it takes to reach a certain number of cases, we solve the exponential decay equation for time. This involves taking logarithms to isolate the time variable, allowing us to determine the number of years needed to reduce cases to the target value.