According to the formula F = (9/5)C + 32, if the temperature in degrees Fahrenheit (F) increases by 27, by how much does the temperature in degrees Celsius (C) increase?
The Correct Answer and Explanation is :
To determine how much the temperature in degrees Celsius (C) increases when the temperature in degrees Fahrenheit (F) increases by 27, we can analyze the given formula:
[
F = \left(\frac{9}{5}\right)C + 32
]
This equation expresses Fahrenheit as a linear function of Celsius. Rearranging to express Celsius in terms of Fahrenheit:
[
C = \frac{5}{9}(F – 32)
]
Step 1: Relationship Between Changes in F and C
The relationship between changes in Fahrenheit ((\Delta F)) and Celsius ((\Delta C)) can be determined by differentiating (F = \left(\frac{9}{5}\right)C + 32) with respect to (C):
[
\frac{dF}{dC} = \frac{9}{5}
]
This indicates that a change of 1 degree Celsius corresponds to a change of ( \frac{9}{5} ) degrees Fahrenheit. Thus, the change in Celsius for a given change in Fahrenheit is:
[
\Delta C = \frac{\Delta F}{\frac{9}{5}} = \Delta F \cdot \frac{5}{9}
]
Step 2: Plugging in the Change in Fahrenheit
Given that (\Delta F = 27), the corresponding change in Celsius is:
[
\Delta C = 27 \cdot \frac{5}{9} = 15
]
Final Answer:
When the temperature in Fahrenheit increases by 27 degrees, the temperature in Celsius increases by 15 degrees.
Explanation (300 words):
The formula (F = \left(\frac{9}{5}\right)C + 32) defines the conversion between Celsius and Fahrenheit scales. The (\frac{9}{5}) factor shows that Fahrenheit changes faster than Celsius due to differing scale intervals. Specifically, a 1-degree change in Celsius corresponds to a (\frac{9}{5})-degree change in Fahrenheit.
When considering an increase in Fahrenheit, the corresponding change in Celsius can be calculated by dividing the Fahrenheit change by (\frac{9}{5}), which is equivalent to multiplying by (\frac{5}{9}). This is because the scales are proportional but have different intervals.
In this problem, the increase of 27 degrees Fahrenheit is scaled back to Celsius using the factor (\frac{5}{9}), resulting in an increase of 15 degrees Celsius. This proportional relationship demonstrates the difference in granularity between the two temperature scales and ensures precise conversions for any temperature change.