Question

Enrico Fermi once pointed out that a standard lecture period $(50\min )$ is close to 1 microcentury. How long is a microcentury in minutes, and what is the percentage difference from Fermi's approximation?

Short answer

A microcentury is approximately 52.596 minutes. The percentage difference from Fermi's approximation is approximately 4.94%.

Step-by-step solution

Calculate the length of a microcentury in minutes.

A microcentury is one millionth of a century. Since a year is approximately 365.25 days (including leap years), a century is 100 years, each day has 24 hours, and each hour has 60 minutes: $$\text{{1 microcentury}}=\frac{100\times 365.25\times 24\times 60}{{10}^6}=\approx 52.596minutes$$

Calculate the percentage difference from Fermi's approximation.

Fermi's approximation was that one lecture period (50 minutes) is close to a microcentury. We can calculate the percentage difference using the formula:$$\text{{percentage difference}}=\frac{\text{{actual value}}-\text{{approximated value}}}{\text{{actual value}}}\times 100$$Substitute the actual value (52.596) and approximated value (50) into the formula:$$\text{{percentage difference}}=\frac{52.596-50}{52.596}\times 100\approx 4.94\mathrm{\%}$$

Key concepts

Time Measurement

Understanding time measurement is crucial for studying various phenomena in science and our daily lives. It involves quantifying the duration between two events using defined units. The most basic units of time are seconds, minutes, hours, days, and years, and these can be combined and converted into larger or smaller units like milliseconds or centuries to fit the context required.

In physics, time measurement is especially important; it helps physicists calculate rates, understand motion, and figure out the timing of events within experiments and natural occurrences. When we talk about a 'microcentury,' we are using a playful unit of time measurement that denotes one millionth of a century. To visualize this, imagine breaking down the 100 years that make up a century into one million equal parts; each part would be a microcentury.

Fermi Approximation

The Fermi approximation is a method of estimation that scientist Enrico Fermi famously used to produce quick, rough calculations. It’s a strategic guessing method that utilizes rounding and simplifying assumptions to arrive at an answer that is close enough to the exact figure for practical purposes. Fermi was known for his exceptional ability to make good approximations with limited information.

These estimations are not randomly made but are based on logical reasoning and basic principles. For instance, when Fermi approximated the length of a lecture to a microcentury, he used the knowledge that a century is 100 years and a typical lecture is about an hour, which is within the same order of magnitude. The technique involves breaking down a complex problem into smaller, more manageable parts, estimating each part, and then combining these estimates to get an overall approximation. This method is beneficial when exact data is unavailable or unnecessary for the level of precision needed in the inquiry.

Percentage Difference Calculation

The percentage difference calculation is a mathematical way to compare two values and determine the extent to which they differ relative to their size. It's expressed as a percentage and is valuable for quantifying the accuracy of estimates or measurements.

Here’s a simple guide on how to calculate it: First, subtract the approximated value from the actual value to determine the difference between them. Then, divide this difference by the actual value, which gives you a decimal. To convert this to a percentage, multiply the result by 100. In our above Fermi approximation example regarding the microcentury, by using the formula $$\text{{percentage difference}}=\frac{\text{{actual value}}-\text{{approximated value}}}{\text{{actual value}}}\times 100$$ with the actual and approximated values provided, we could find that the approximate value of a 50-minute lecture period is close but not exact, with a percentage difference of approximately 4.94%.

This calculation helps us quantify the accuracy of Fermi's estimation and is widely used in many fields such as finance, statistics, and experimental science, to assess the reliability of results and models.