Optimizing Bulb Replacement Frequency for Minimizing Burnouts

[FaradayLaws]

Homework Statement

A manufacturing plant utilizes 3000 electric light bulbs whose length of life is normally distributed
with mean 500 hours and standard deviation 50 hours. To minimize the number of
bulbs that burn out during operating hours, all the bulbs are replaced after a given period of
operation. How often should the bulbs be replaced if we want not more than 1% of the bulbs
to burn out between replacement periods?

Homework Equations

The Attempt at a Solution

I'm having great difficulty with this question. I'm confused as to how I can find how often the bulbs should be replaced (within what times interval?) I was wondering if this maybe is a combination of a normal distribution with an expontial ? I was thinking of finding standizing the distribution
P(Z <= x-500/50)= 0.01 and then simply solving for x. But I am not sure how that even helps me. Overall I'm just quite confused with this problem.

Thank you !

 

[hgfalling]

Suppose we pick one bulb out of the 3000, and replace it after time t. What is the probability that the bulb will be burned out at time t?

Now suppose that we use the same time t for all the bulbs. What percentage of the bulbs will be burned out at time t? (Hint: it's the same as the first question)

Our answer should be a function of t. Then we just find the value of t such that f(t) = 0.01.