How do you calculate the reliability for a different time period, say 2000 hours?

To calculate the reliability \( R(t) \) for a different time period, such as 2000 hours, you can use the following steps:

### Formula for Reliability
The reliability function \( R(t) \) is given by:

\[
R(t) = e^{-\lambda t}
\]

Where:
- \( \lambda \) is the failure rate (in failures per hour).
- \( t \) is the time period for which you want to calculate the reliability.

### Step-by-Step Calculation

1. **Determine the failure rate (\( \lambda \)):**
   From the previous calculation, we found:
   \[
   \lambda \approx 1.98 \times 10^{-6} \text{ failures per hour}
   \]

2. **Choose the time period \( t \):**
   For this example, the time period is \( t = 2000 \) hours.

3. **Substitute the values into the reliability formula:**
   \[
   R(2000) = e^{-\lambda \times 2000} = e^{-1.98 \times 10^{-6} \times 2000}
   \]

4. **Calculate the exponent:**
   \[
   \text{Exponent} = -1.98 \times 10^{-6} \times 2000 = -0.00396
   \]

5. **Compute the reliability \( R(2000) \):**
   \[
   R(2000) = e^{-0.00396} \approx 0.99605
   \]

### Final Result:
The reliability for a period of 2000 hours is approximately **0.99605**.

This means there is approximately a 99.6% chance that an item will not fail within 2000 hours.

### Do you want more details or have any other questions?

---

Here are some related questions:
1. How would the reliability change if the time period is increased to 10,000 hours?
2. What does the reliability tell you about the expected performance of the items?
3. How does the reliability compare for shorter time periods like 500 hours?
4. Can reliability be calculated for a system with varying failure rates over time?
5. How can you improve the reliability of a system?

**Tip:** For long-term reliability predictions, consider factors like wear-out mechanisms, which may cause the failure rate to increase over time.

Learn how to calculate the reliability for a time period of 2000 hours using the reliability function. Follow a detailed step-by-step guide with formulas and calculations. Gain insights into the probability of an item not failing within 2000 hours.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation