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−ct, where c is a positive constant. How does the value of c affect the reliability of a chip?
as c gets large, e^-ct decreases, meaning F approaches 1.
That is, as c grows, so does the fraction likely to fail.
Actually, of course, c does not actually affect the reliability. It just serves as a predictor of how many can be expected to fail.
To understand how the value of the constant "c" affects the reliability of a computer chip, we need to analyze the given formula: F = 1 - e^(-ct), where F represents the fraction of chips that will fail after t years of service, and c is a positive constant.
In this formula, e is a mathematical constant known as Euler's number or the base of the natural logarithm. It is approximately equal to 2.71828. The exponential function e^(-ct) describes the decay of chip reliability with time.
The value of "c" determines the rate at which the fraction F decreases over time. The larger the value of "c," the faster the decay or failure rate of the chips.
Here are some observations:
1. Larger "c" values: When "c" is larger, the exponential term e^(-ct) decreases more rapidly as time increases. Consequently, the fraction F decreases faster, indicating a higher rate of chip failures. This means that chips with larger "c" values are less reliable since they are more likely to fail sooner.
2. Smaller "c" values: Conversely, when "c" is smaller, the exponential term e^(-ct) decreases more slowly over time. As a result, the fraction F decreases at a slower rate, indicating a lower rate of chip failures. Chips with smaller "c" values are considered more reliable since they are less likely to fail in the given time frame.
To summarize, the value of the constant "c" in the formula F = 1 - e^(-ct) determines the rate at which a computer chip's reliability declines over time. A larger "c" value corresponds to a faster rate of failure, while a smaller "c" value indicates a slower rate of failure or higher reliability. Manufacturers must carefully choose the value of "c" to meet the desired reliability requirements for their computer chips.