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Chapter 4: Problem 82

Computer chips For manufacturers of computer chips, it is important to consider the fraction \(F\) of chips that will fail after \(t\) years of service. This fraction can sometimes be approximated by the formula \(F=1-e^{-c t},\) where \(c\) is a positive constant. (a) How does the value of \(c\) affect the reliability of a chip? (b) If \(c=0.125,\) after how many years will \(35 \%\) of the chips have failed?

Short Answer

Expert verified
(a) A larger \( c \) leads to faster failure, thus lower reliability. (b) Approximately 3.45 years.

Step by step solution

01

Understanding the Effect of c on Reliability

The formula for the fraction of chips that fail is given by \( F = 1 - e^{-ct} \). Here, \( c \) is a constant that affects the rate at which chips fail. If \( c \) is larger, \( e^{-ct} \) decreases faster, leading to a quicker increase in \( F \). Thus, a higher \( c \) means chips are less reliable over the same time period, as they fail more quickly. Conversely, a smaller \( c \) implies better reliability as chips fail more slowly.
02

Setting Up the Equation for Part (b)

We want to find the time \( t \) when 35% of the chips fail, meaning \( F = 0.35 \). The equation becomes \( 0.35 = 1 - e^{-0.125t} \). Rearrange to solve for \( e^{-0.125t} \), yielding \( e^{-0.125t} = 1 - 0.35 = 0.65 \).
03

Solving for t

Take the natural logarithm of both sides to isolate \( t \): \( \ln(e^{-0.125t}) = \ln(0.65) \). Simplify using the property \( \ln(e^x) = x \), resulting in \( -0.125t = \ln(0.65) \). Solve for \( t \) by dividing both sides by \(-0.125\): \( t = \frac{\ln(0.65)}{-0.125} \).
04

Calculating t

Compute the value of \( t \). Use a calculator to find \( \ln(0.65) \approx -0.4308 \). Thus, \( t = \frac{-0.4308}{-0.125} \approx 3.4464 \). Round to an appropriate number of decimal places if necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reliability Engineering
Reliability engineering is a field focused on ensuring that products are designed and manufactured to perform consistently over their intended lifespan. This concept is crucial in industries like computer chip manufacturing, where reliability requirements are exceptionally high. Manufacturers use mathematical models to predict how long a product will last before it fails.

In the context of computer chips, reliability can be described using the formula for exponential decay: \( F = 1 - e^{-ct} \). This formula helps predict the fraction of chips that may fail after a particular period, where \( c \) represents the rate of failure. Essentially, if you imagine a batch of chips, the formula allows you to calculate how many will still be functioning after a certain amount of time.
  • A larger \( c \) means a faster failure rate and lower reliability.
  • A smaller \( c \) suggests a slower failure rate and higher reliability.
Reliability engineering uses these concepts to help improve product designs, processes, and testing procedures to extend product life and satisfy consumer expectations.
Natural Logarithm
The natural logarithm, often denoted as \( \ln \), is a fundamental mathematical function that is commonly used in modeling exponential growth or decay, such as in reliability analysis or population studies. The natural logarithm has a base of \( e \), which is approximately 2.71828.

In the context of this exercise, when solving for the time \( t \) in the equation \( 0.35 = 1 - e^{-0.125t} \), the natural logarithm comes into play. By taking the natural logarithm of both sides of the equation, you can move the decay component \( e^{-0.125t} \) out of the exponent:
  • Start with \( e^{-0.125t} = 0.65 \)
  • Taking \( \ln \) of both sides gives \( \ln(e^{-0.125t}) = \ln(0.65) \)
  • Simplify using \( \ln(e^x) = x \) to get \( -0.125t \) on one side
The natural logarithm allows us to linearize the equation for easy solving, emphasizing its utility in various scientific and engineering applications.
Failure Rate Analysis
Failure rate analysis is an essential component of reliability engineering, helping to determine how often failures occur in a product during its life cycle. In this exercise, the parameter \( c \) in the formula for the fraction of chips that fail, \( F = 1 - e^{-ct} \), is directly related to the failure rate. The higher the value of \( c \), the greater the likelihood of failure over a given period.

  • High \( c \) value: Indicates rapid failure rate, implying chips become unreliable quickly.
  • Low \( c \) value: Suggests slow failure rate, which is desirable for reliability.
Failure rate analysis is not only useful for understanding and predicting when failures might happen, but it also aids manufacturers in improving product quality by identifying weaknesses in design or production. Through this analysis, engineers can develop strategies to mitigate risks and enhance the longevity and robustness of their products.

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