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Think about what the "optimal length of time" to own this means. It's the time at which it becomes cheaper to replace it than to repair it.
The replace cost is 10000. The repair cost is 100t^(7/4). When does 100t^(7/4) equal 10000?
t^(7/4) = 100.
t = 100^(4/7)
t = 13.894. The optimal length of time to own the machine is 13.894 years. At that point, it's cheaper to replace it than repair it.
You spend $10,000 every 13.894 years, plus $500 a year for operating expenses. The annual cost should be easy enough for you to find.
