Calculate Probability Machine Part Lifetime < 6: Integration Problem

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In summary, the lifetime of a machine part has a continuous distribution on the interval (0,40) with probability density function f, where f(x) is proportional to (10 + x)^(-2). To calculate the probability that the lifetime of the machine part is less than 6, we need to put a constant k in the equation since f(x) is proportional to (10 + x). This allows us to set up the integration to find the proportionality constant and ultimately calculate the desired probability.
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Homework Statement

the lifetime of a machine part has a continuous distribution on the interval(0,40) with probability density function f, where f(x) is proportional to (10 + x)^(-2) Calculate the probability that the lifetime of the machine part is less than 6.



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The Attempt at a Solution

My book says that since f(x) is proportional to (10 + x), f(x)=k(10+x)^(-2). My question is why we need to put the k in the equation? Is it merely because the question says that they're proportional and so we need to set up that integration to find by how much they're proportional?
 
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Yes.
 

1. What is the purpose of calculating the probability of a machine part having a lifetime less than 6 years?

The purpose of calculating this probability is to determine the likelihood of a machine part failing within 6 years. This information can be used to assess the reliability of the part and make informed decisions about maintenance and replacement schedules.

2. How is the probability of a machine part's lifetime calculated?

The probability of a machine part's lifetime can be calculated using statistical methods, such as the Poisson distribution or the exponential distribution. It involves analyzing data on the failure times of similar machine parts and using mathematical formulas to estimate the likelihood of failure within a given time frame.

3. What factors can affect the accuracy of the calculated probability?

Some factors that can affect the accuracy of the calculated probability include the quality and completeness of the data used, the assumptions made in the mathematical model, and the variability of the environment and conditions in which the machine part operates.

4. Can the calculated probability be used to predict the exact lifetime of a machine part?

No, the calculated probability only provides an estimate of the likelihood of the machine part failing within a certain time frame. It cannot predict the exact lifetime of the part, as there are many variables that can affect its longevity.

5. How can the calculated probability be used in decision-making for a machine part's maintenance or replacement?

The calculated probability can be used to inform decisions about the maintenance or replacement of the machine part. For example, if the probability is high, it may be advisable to increase the frequency of maintenance checks or consider replacing the part sooner rather than later.

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