At the Amundsen-Scott South Pole base station in Antarctica, when the temperature is \(-100.0^{\circ} \mathrm{F}\) , researchers who live there can join the 4300 \(\mathrm{Club}^{\prime \prime}\) by stepping into a sauna heated to \(200.0^{\circ} \mathrm{F}\) then quickly running outside and around the pole that marks the South Pole. What are these temperatures in $^{\circ} \mathrm{C} ?\( What are these temperatures in \)\mathrm{K}$ ? If you measured the temperatures only in \(^{\circ} \mathrm{C}\) and \(\mathrm{K}\) , can you become a member of the \(" 300 \mathrm{Club}^{\prime \prime}\) (that is, is there a 300 .-degree difference between the temperature extremes when measured in \(^{\circ} \mathrm{C}\) and \(\mathrm{K}\) )?

Step by step solution

01

Convert Fahrenheit to Celsius

To convert Fahrenheit to Celsius, we use the formula: \[Celsius = \frac{5}{9} (Fahrenheit - 32)\] We are given two Fahrenheit temperatures: 1. -100.0°F 2. 200.0°F Let's convert these to Celsius: 1. \(Celsius_1 = \frac{5}{9} (-100.0 - 32)\) 2. \(Celsius_2 = \frac{5}{9} (200.0 - 32)\)

02

Calculate Celsius Temperatures

Calculate the Celsius temperatures using the converted temperatures formula: 1. \(Celsius_1 = \frac{5}{9} (-132) = -73.3°C\) 2. \(Celsius_2 = \frac{5}{9} (168) = 93.3°C\)

03

Convert Celsius to Kelvin

To convert Celsius to Kelvin, we use the formula: \[Kelvin = Celsius + 273.15\] We have two Celsius temperatures: 1. -73.3°C 2. 93.3°C Let's convert these to Kelvin: 1. \(Kelvin_1 = -73.3 + 273.15\) 2. \(Kelvin_2 = 93.3 + 273.15\)

04

Calculate Kelvin Temperatures

Calculate the Kelvin temperatures using the converted temperatures formula: 1. \(Kelvin_1 = 199.85 K\) 2. \(Kelvin_2 = 366.45 K\)

05

Check for a 300-degree difference

To see if a researcher can become a member of the 300 Club, we need to check if the difference in the temperature extremes is 300 when measured in Celsius and Kelvin: 1. Celsius Difference: |93.3 - (-73.3)| = 166.6 2. Kelvin Difference: |366.45 - 199.85| = 166.6 The difference between the temperature extremes is the same in Celsius and Kelvin (166.6), but it is not 300. Therefore, a researcher cannot become a member of the 300 Club when measuring temperatures only in Celsius and Kelvin.

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