A Web server has five major components that must all function in order for it to operate as intended. Assuming that each component of the system has the same reliability and one of the components will have a backup with a reliability equal to that of any one of the other components, what is the minimum reliability each one must have in order for the overall system to have a reliability of 0.696

The answer is below

Explanation:

Let x represent the reliability of each of the five major components. Since all the components must function before the system operates, hence they are in series with each other.

One component has a backup of the same reliability, the reliability of the back up system = 1 - (1 - x)² = 2x - x²

Therefore the reliability of the system is:

x * x * x * x * (2x - x²) = 0.696

2x⁵ - x⁶ = 0.696

x⁶ - 2x⁵ + 0.696 = 0

Solving the polynomial equation online using byjus.com gives:

x = 0.915038

x = 1.97695

x = -0.632585 - 0.429075 i

x = -0.632585 + 0.429075 i

x = 0.186590 - 0.789741 i

x = 0.186590 + 0.789741 i

Since the reliability is not a complex number and it is less than or equal to 1, hence:

x = 0.915038, x ≅ 0.915