C = 5 9 ( F − 32 ) The equation above shows how temperature F , measured in degrees Fahrenheit, relates to a temperature C , measured in degrees Celsius. Based on the equation, which of the following must be true? A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 5 9 degree Celsius. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit. A temperature increase of 5 9 degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius. A) I only B) II only C) III only D) I and II only

To solve this problem, we need to analyze the relationship between temperature changes in Fahrenheit and Celsius using the given formula:

$C = \frac{5}{9} (F - 32)$

Break Down the Problem

1. Determine how a change in Fahrenheit affects Celsius.
2. Determine how a change in Celsius affects Fahrenheit.

Relevant Concepts

1. Fahrenheit to Celsius Conversion: The formula $C = \frac{5}{9} (F - 32)$ can be used to find how changes in Fahrenheit translate to changes in Celsius.
2. Celsius to Fahrenheit Conversion: The inverse of the formula can be used to find how changes in Celsius translate to changes in Fahrenheit.

Analysis and Detail

1. Change in Fahrenheit:

   • If $F$ increases by 1 degree, the change in Celsius ($\Delta C$) can be calculated as: $\Delta C = \frac{5}{9} \times 1 = \frac{5}{9}$
   • This confirms that a temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius.

2. Change in Celsius:

   • To find how a change in Celsius affects Fahrenheit, we rearrange the formula to solve for $F$: $F = \frac{9}{5}C + 32$
   • If $C$ increases by 1 degree, the change in Fahrenheit ($\Delta F$) can be calculated as: $\Delta F = \frac{9}{5} \times 1 = 1.8$
   • This confirms that a temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

Verify and Summarize

• Statement I: A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius. True.
• Statement II: A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit. True.
• Statement III: A temperature increase of $\frac{5}{9}$ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius. False.

Final Answer

The correct answer is $\boxed{D}$ (I and II only).