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Chapter 1: Problem 26

A cold front moves through and the temperature drops by 20 degrees. In which temperature scale would this 20 degree change represent the largest change in temperature?

Short Answer

Expert verified
The largest change in temperature when the temperature drops by 20 degrees would be in the Fahrenheit scale, as a 20-degree drop in Celsius corresponds to a 36-degree drop in Fahrenheit.

Step by step solution

01

Understanding the relationship between Celsius and Fahrenheit temperatures

The formula to convert Celsius to Fahrenheit is given by: \(F = \frac{9}{5}C + 32\) In this exercise, we need to compare the change in temperature in both Celsius and Fahrenheit. So we need to find the relationship between the changes in Celsius and Fahrenheit.
02

Calculating the change in Fahrenheit when the temperature in Celsius drops by 20 degrees

Suppose that the temperature in Celsius drops by 20 degrees, from, for example, an initial temperature of \(C_1 = 20\) degrees Celsius to a final temperature of \(C_2 = 0\) degrees Celsius. Using the Celsius to Fahrenheit conversion formula, we can find the corresponding initial and final Fahrenheit temperatures: Initial Fahrenheit temperature: \(F_1 = \frac{9}{5}(20) + 32\) Final Fahrenheit temperature: \(F_2 = \frac{9}{5}(0) + 32\) Now we need to find the difference between the initial and final Fahrenheit temperatures as follows: Fahrenheit difference: \(F_2 - F_1 = (\frac{9}{5}(0) + 32) - (\frac{9}{5}(20) + 32)\)
03

Simplifying the Fahrenheit difference equation

Now we need to simplify the equation to get the numerical value of the Fahrenheit difference: \(F_2 - F_1 = -\frac{9}{5}(20)\)
04

Solving for the Fahrenheit difference

Finally, we can solve for the Fahrenheit difference: \(F_2 - F_1 = -36\) This means that when the temperature drops by 20 degrees Celsius, it drops by 36 degrees Fahrenheit.
05

Comparing the changes in Celsius and Fahrenheit

Since the temperature change in Celsius is 20 degrees and the corresponding temperature change in Fahrenheit is 36 degrees, we can see that the 20-degree drop in the Celsius scale represents a larger change in the Fahrenheit scale. Thus, the largest change in temperature when the temperature drops by 20 degrees would be in the Fahrenheit scale.

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