Final answer:
To find the 75th percentile of lifetimes for a mechanical component with an exponential distribution, you can use the cumulative distribution function (CDF). The 75th percentile is approximately 6.089 years.
Step-by-step explanation:
To find the 75th percentile of lifetimes for a mechanical component with an exponential distribution, we need to find the value, x, such that P(T <= x) = 0.75. In this case, the mean lifetime is 8.4 years.
We can use the formula for the cumulative distribution function (CDF) of the exponential distribution to solve for x:
P(T <= x) = 1 - e^(-λx)
Where λ = 1/mean. Substituting the given mean, we have:
P(T <= x) = 1 - e^(-1/(8.4)x)
Solving for x when P(T <= x) = 0.75, we find:
0.75 = 1 - e^(-1/(8.4)x)
e^(-1/(8.4)x) = 0.25
-1/(8.4)x = ln(0.25)
x = (-8.4) * ln(0.25) ≈ 6.089
Therefore, the 75th percentile of lifetimes for this component is approximately 6.089 years.