Question

There are a number of identical components in a parallel system. When the system reliability is 0.97 and the reliability of each individual component is 0.68, the number of identical components in the system is (if actual value is a fraction, it may be rounded up to the next higher integer).

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

In parallel systems, the reliability increases as more components are added. Use the formula to determine the number of components required for a given system reliability.

Answer 1

The Correct Option is B

For a parallel system, the reliability \( R_{\text{sys}} \) is given by: \[ R_{\text{sys}} = 1 - \prod_{i=1}^{n} (1 - R_i) \] where \( R_i \) is the reliability of each individual component, and \( n \) is the number of components in parallel. Here, \( R_{\text{sys}} = 0.97 \) and \( R_i = 0.68 \). Substituting these values: \[ 0.97 = 1 - (1 - 0.68)^n \] \[ 0.03 = (0.32)^n \] Taking the natural logarithm of both sides: \[ \ln(0.03) = n \ln(0.32) \] \[ n = \frac{\ln(0.03)}{\ln(0.32)} \approx 3.44. \] Rounding up, we get \( n = 4 \). Thus, the number of components in the system is (B) 4.