Problem 66 - Calculus for Biology and Medicine

Question

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having failed by age \(x_{m}\) is \(0.5\). Use a graphing calculator to numerically approximate the median lifetime if the hazard-rate function is $$ \lambda(x)=0.5+0.1 e^{0.2 x}, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The median lifetime is approximately 1.83.

Step 1: Understand the problem

The median lifetime, denoted as \(x_m\), is the age at which there is a 50% probability of the system not having failed. We need to find \(x_m\) such that the cumulative survival probability is 0.5.

Step 2: Define the survival function

The survival function \(S(x)\) is related to the hazard-rate function \(\lambda(x)\) by the equation \(S(x) = e^{- ext{integral}(\lambda(x), dx)}\). Our task is to evaluate the integral of \(\lambda(x)\) and then solve for \(S(x_m) = 0.5\).

Step 3: Integrate the hazard function

Compute the integral of the given \(\lambda(x) = 0.5 + 0.1 e^{0.2x}\). This results in \(\int \lambda(x) \, dx = 0.5x + 0.1 \cdot \frac{e^{0.2x}}{0.2}\). Simplifying gives \(0.5x + 0.5e^{0.2x}\).

Step 4: Formulate the survival function

From the integration, the survival function becomes \(S(x) = e^{-0.5x - 0.5 e^{0.2x}}\). Set \(S(x_m) = 0.5\) to find \(x_m\).

Step 5: Solve the equation for the median lifetime

We need to solve \(0.5 = e^{-0.5x_m - 0.5e^{0.2x_m}}\). Taking the natural logarithm of both sides results in solving \(-0.5x_m - 0.5e^{0.2x_m} = \ln(0.5)\).

Step 6: Numerically approximate the solution

Using a graphing calculator, input the function \(-0.5x - 0.5e^{0.2x}\) and find the value of \(x\) that satisfies \(-0.5x - 0.5e^{0.2x} = \ln(0.5)\). The approximation gives \(x_m \approx 1.83\).

Key Concepts

Hazard-Rate Function

The hazard-rate function, often denoted by \( \lambda(x) \), plays a critical role in reliability engineering and survival analysis. It represents the rate at which a failure occurs in a system at any given time, given that the system has survived up to that time. This function is crucial because it provides insights into the failure dynamics of a system.

• For the given problem, the hazard-rate function is: \( \lambda(x) = 0.5 + 0.1 e^{0.2 x} \).
• \( x \) is the time, starting from zero, where the system has not failed.
• The function implies the failure rate increases over time due to the exponential component \( 0.1 e^{0.2x} \).

Understanding how the hazard rate evolves makes it possible to predict and mitigate potential system failures. It allows engineers to plan maintenance activities strategically.

Survival Function

The survival function, often represented as \( S(x) \), describes the probability that a system or component will continue to function up to a certain time \( x \). In mathematical terms, it is related to the hazard-rate function \( \lambda(x) \) through an exponential function.

• The relationship is \( S(x) = e^{- \text{Integral}(\lambda(x), dx)} \).
• For the given hazard-rate function \( \lambda(x) = 0.5 + 0.1 e^{0.2x} \), the survival function becomes \( S(x) = e^{-0.5x - 0.5e^{0.2x}} \).

This function essentially works as a counterpart to the hazard-rate, providing a complete picture of reliability over time. In this problem, the objective is to solve for \( x_m \) such that \( S(x_m) = 0.5 \), indicating a 50% survival probability.

Numerical Approximation

Numerical approximation involves using computational methods to find an approximate solution to mathematical problems that cannot be easily solved analytically. Due to the complexity of some equations, such as those including exponential terms, numerical methods become very useful.

• For this problem, we require a numerical solution to the equation: \(-0.5x - 0.5e^{0.2x} = \ln(0.5)\).
• This procedure involves using a graphing calculator or numerical software to iteratively test values of \( x \) to find where this equation holds true.
• This allows us to approximate the median lifetime as \( x_m \approx 1.83 \).

Numerical methods are instrumental in solving real-world problems where exact solutions are challenging to extract, thus offering feasible and practical insights.

Integration

Integration is a fundamental concept in calculus, used to calculate areas under curves, among other things. In reliability analysis, integration is employed to derive the survival function from the hazard-rate function.

• In this context, the integral of the hazard-rate function is computed as \( \int \lambda(x) \, dx = 0.5x + 0.5e^{0.2x} \).
• This result is vital as it helps form the exponential component \( e^{-0.5x - 0.5e^{0.2x}} \) in the survival function.

Calculating integrals allows us to understand the aggregate effect of the hazard-rate over time, forming the basis for evaluating probabilities in survival contexts. Proficiency in integration helps analyze dynamic systems and predict their longevity effectively.