Question

A thermometer gives a reading of \(96.1^{\circ} \mathrm{F} \pm 0.2^{\circ} \mathrm{F}\) . What is the temperature in \(^{\circ} \mathrm{C} ?\) What is the uncertainty?

Short answer

The temperature in Celsius is \(35.61^{\circ}\mathrm{C} \pm 0.11^{\circ}\mathrm{C}\).

Step-by-step solution

Identify the conversion formula

We need to use the conversion formula between Fahrenheit and Celsius, which is given by: \[C = \frac{5}{9}(F - 32)\] Where \(C\) is the temperature in Celsius, and \(F\) is the temperature in Fahrenheit.

Convert the given temperature

Using the formula, replace the \(F\) with the given Fahrenheit temperature \(96.1^{\circ}\mathrm{F}\): \(C = \frac{5}{9}(96.1 - 32)\) Now, calculate the result: \(C = \frac{5}{9}(64.1) = 35.61^{\circ}\mathrm{C}\) So, the temperature in Celsius is \(35.61^{\circ}\mathrm{C}\).

Convert the uncertainty

To determine the uncertainty in Celsius, treat the uncertainty as a separate temperature and use the same conversion formula: \(\Delta C = \frac{5}{9}(- 0.2^{\circ}\mathrm{F})\) Calculate the result: \(\Delta C = \frac{5}{9}(-0.2) = -0.11^{\circ}\mathrm{C}\) By convention, we always express the uncertainty as a positive value, so the uncertainty in Celsius is \(0.11^{\circ}\mathrm{C}\).

Write the final answer

Combine the converted temperature and its uncertainty in Celsius: The temperature in Celsius is \(35.61^{\circ}\mathrm{C} \pm 0.11^{\circ}\mathrm{C}\).

Key concepts

Fahrenheit to Celsius conversion

Converting temperatures between Fahrenheit and Celsius is a common task in everyday life and scientific calculations. The formula that bridges the temperature scales of Fahrenheit and Celsius is:

• \[ C = \frac{5}{9}(F - 32) \]

where \(C\) is the temperature in Celsius, and \(F\) is the temperature in Fahrenheit. This formula helps you easily convert Fahrenheit readings into Celsius by adjusting for the freezing point difference and the scale interval per degree.
To apply this formula effectively, start by subtracting 32 from the Fahrenheit value, which aligns with the difference in the freezing points of water between the two scales. Then multiply by the fraction \(\frac{5}{9}\), which adjusts the Fahrenheit scale interval per degree to match that of Celsius, which is smaller.
This conversion comes in handy whether it's determining the day's temperature in a more familiar scale or if you are working on scientific experiments that require precise metrics.

Uncertainty calculation

In scientific measurements, it's crucial to account for uncertainty, indicating the range within which the true value is expected to lie. When converting temperatures, uncertainty ensures the accuracy and reliability of the measurement.
In our context, the uncertainty of the thermometer reading in Fahrenheit is \(\pm 0.2^{\circ} \mathrm{F}\). To convert this into Celsius, apply the Fahrenheit to Celsius conversion formula to the uncertainty value:

• \[ \Delta C = \frac{5}{9}(\pm 0.2^{\circ}\mathrm{F}) \]

After calculation, this depends only on the numerical conversion of 0.2 from Fahrenheit to Celsius, resulting in \(\pm 0.11^{\circ} \mathrm{C}\).
Always remember to present uncertainty as a positive number as it reflects the magnitude of the possible deviation, not the direction. Understanding how to convert and interpret uncertainty is essential for ensuring that scientific findings are not only correct but also consistent and credible.

Thermometer reading precision

Thermometer reading precision relates to how finely a thermometer can measure, typically noted by the smallest division on its scale. Precision in a thermometer reading indicates how consistently readings can be duplicated, a vital observation for both practical usage and scientific analysis.
In the exercise, a thermometer measured a temperature of \(96.1^{\circ} \mathrm{F}\) with a certainty of \(\pm 0.2^{\circ} \mathrm{F}\). The given uncertainty highlights the precision limit of the thermometer.
Thermometers come in varying precision levels, influenced by their design and intended use. Higher precision thermometers, such as digital ones, may offer more subdivisions and lesser uncertainty. Understanding the precision helps users select appropriate thermometers for their specific needs, guaranteeing sufficiently accurate and reliable readings.
Practical applications range from common household tasks like setting a thermostat correctly, to critical scientific activities where precision is of utmost importance. This allows you to appreciate not just the temperature reading itself, but the confidence interval around it, which further informs any temperature-dependent decisions or adjustments you might have to make.