Verified Answer :
- Convert the Fahrenheit value to Celsius: \(T(^{\circ}\text{C}) = (92. - 32) \times \frac{5}{9} \approx 33.^{\circ}\text{C}\) (rounded to two significant figures).
- Add 273.15 to the Celsius temperature to find the value in Kelvin: \(33. + 273.15 = 306.15\).
- Round the result to the nearest whole number to maintain the precision of the original Celsius measurement: \(\boxed{306. \text{ K}}\).
Explanation
- Converting Fahrenheit to Celsius
The conversion from Fahrenheit to Kelvin involves two primary steps: converting the Fahrenheit temperature to Celsius, and then converting the Celsius result to Kelvin. The given temperature is \(92.^{\circ}\text{F}\), which has two significant figures because the decimal point signifies that the last digit is measured precisely. The conversion formula from Fahrenheit to Celsius is \(T(^{\circ}\text{C}) = (T(^{\circ}\text{F}) - 32) \times \frac{5}{9}\). Since \(32\) is an exact number in this definition, we first perform the subtraction \(92. - 32 = 60.\). This result \(60.\) retains the precision of the decimal point, maintaining two significant figures. Multiplying this by the fraction \(\frac{5}{9}\), we obtain approximately \(33.333...\) degrees Celsius. Based on the rule for multiplication and division, we retain only the number of significant figures present in the least precise factor, which is two. Thus, we round the result to \(33.^{\circ}\text{C}\).
- Converting Celsius to Kelvin
Next, we convert the Celsius temperature to Kelvin using the formula \(T(\text{K}) = T(^{\circ}\text{C}) + 273.15\). Substituting our value, we have \(33. + 273.15 = 306.15\). According to the rules of significant figures for addition and subtraction, the final result must be rounded to the same degree of precision as the measurement with the fewest decimal places. Since \(33.^{\circ}\text{C}\) is precise to the units place (the decimal point indicates it is measured to the ones place), our sum must also be rounded to the ones place. Therefore, \(306.15\) is rounded to \(306.\) K.
Examples
Understanding temperature conversions with significant figures is essential in scientific and meteorological reporting. For instance, when weather stations report temperatures, they must ensure accuracy and consistent precision so that data can be correctly interpreted in international research. Whether working with industrial heating systems or analyzing environmental data, adhering to these rules helps maintain the integrity of scientific measurements and prevents misleading conclusions about precision.