Final answer:
To find the fraction of bearings that would fail in six months, use the cumulative distribution function (CDF) of the Weibull distribution with the given parameters. To find the length of time to limit the design life to no more than 1% failure, solve the equation P(X < t) = 0.01.
Step-by-step explanation:
To answer this question, we need to use the Weibull distribution and the given parameters. The Weibull distribution is a continuous probability distribution that is commonly used to model failure times of mechanical or electronic systems.
a) Fraction of bearings that would fail in six months:
To find the fraction of bearings that would fail in six months, we need to calculate the probability that a bearing fails before six months. We can use the cumulative distribution function (CDF) of the Weibull distribution with the given parameters.
Given m = 2 and L₁₀ = 1 year, we can derive the scale parameter (λ) using the formula: L₁₀ = λ * (21/m - 1)
Solving for λ, we get:
λ = L₁₀ / (21/m - 1) = 1 / (21/2 - 1) = 1 / (2 - 1) = 1 year
Now, we can calculate the probability of failure before six months:
P(X < 0.5) = 1 - exp(-(0.5/λ)m)
= 1 - exp(-((0.5/1)2)
= 1 - exp(-0.25)
= 1 - 0.7788
= 0.2212
Therefore, the fraction of bearings that would fail in six months is approximately 0.2212 or 22.12%.
b) Length of time to limit the design life to no more than 1% failure:
To find the length of time to guarantee no more than 1% failure, we need to find the time (t) at which the cumulative distribution function (CDF) of the Weibull distribution is equal to 0.01. We can solve this equation:
P(X < t) = 0.01
1 - exp(-(t/λ)m) = 0.01
exp(-(t/λ)m) = 0.99
-(t/λ)m = ln(0.99)
t = -λ * (ln(0.99))1/m
t = -1 * (ln(0.99))1/2
t ≈ 0.0259 years
Therefore, to guarantee no more than 1% failure, the length of time to limit the design life is approximately 0.0259 years or 9.45 days.