If C is the temperature in degrees Celsius and F is the temperature in degrees Fahrenheit, then the relationship between...

1. Translate the problem requirements: We need to find how much temperature differs on the Celsius scale when the difference on Fahrenheit scale is 45 degrees. The key insight is that we're looking for the difference between temperatures, not the actual temperatures themselves.
2. Establish the relationship for temperature differences: Use the given conversion equation \(9\mathrm{C} = 5(\mathrm{F} - 32)\) to derive how differences in Fahrenheit relate to differences in Celsius.
3. Apply the difference relationship: Since we know the Fahrenheit difference is 45 degrees, calculate the corresponding Celsius difference using the derived relationship.

Execution of Strategic Approach

1. Translate the problem requirements

Let's think about what this problem is really asking. We have two thermometers - one reading in Fahrenheit and one in Celsius. On a particular day, if we look at the hottest and coldest temperatures recorded, the difference between these extremes was 45 degrees when measured in Fahrenheit.

For example, maybe it was \(80°\mathrm{F}\) at the hottest point and \(35°\mathrm{F}\) at the coldest point of the day. The difference would be \(80 - 35 = 45\) degrees Fahrenheit.

Now we want to know: what would that same temperature difference be if we measured it in Celsius instead?

The key insight here is that we're not converting specific temperatures - we're finding how much a difference of \(45°\mathrm{F}\) equals in Celsius degrees.

Process Skill: TRANSLATE

2. Establish the relationship for temperature differences

Let's work with the conversion formula we're given: \(9\mathrm{C} = 5(\mathrm{F} - 32)\)

Now, here's the clever part. If we have two temperatures, let's call them \(\mathrm{F}_1\) and \(\mathrm{F}_2\) in Fahrenheit, and their corresponding Celsius temperatures \(\mathrm{C}_1\) and \(\mathrm{C}_2\), then:

• For the first temperature: \(9\mathrm{C}_1 = 5(\mathrm{F}_1 - 32)\)
• For the second temperature: \(9\mathrm{C}_2 = 5(\mathrm{F}_2 - 32)\)

To find the relationship between the differences, let's subtract the second equation from the first:
\(9\mathrm{C}_1 - 9\mathrm{C}_2 = 5(\mathrm{F}_1 - 32) - 5(\mathrm{F}_2 - 32)\)

Simplifying the right side:
\(9\mathrm{C}_1 - 9\mathrm{C}_2 = 5\mathrm{F}_1 - 160 - 5\mathrm{F}_2 + 160\)
\(9\mathrm{C}_1 - 9\mathrm{C}_2 = 5\mathrm{F}_1 - 5\mathrm{F}_2\)
\(9(\mathrm{C}_1 - \mathrm{C}_2) = 5(\mathrm{F}_1 - \mathrm{F}_2)\)

This tells us that the difference in Celsius \((\mathrm{C}_1 - \mathrm{C}_2)\) relates to the difference in Fahrenheit \((\mathrm{F}_1 - \mathrm{F}_2)\) by:
\(9 \times \text{(Celsius difference)} = 5 \times \text{(Fahrenheit difference)}\)

3. Apply the difference relationship

Now we can use our relationship. We know the Fahrenheit difference is 45 degrees, so:
\(9 \times \text{(Celsius difference)} = 5 \times 45\)
\(9 \times \text{(Celsius difference)} = 225\)

To find the Celsius difference:
\(\text{Celsius difference} = 225 \div 9 = 25\)

Let's verify this makes sense: A 45-degree difference in Fahrenheit corresponds to a 25-degree difference in Celsius. This seems reasonable since Celsius degrees are "bigger" than Fahrenheit degrees (each Celsius degree represents a larger temperature change).

4. Final Answer

The temperature extremes differed by 25 degrees on the Celsius scale.

Looking at our answer choices, this corresponds to choice C. 25.

Common Faltering Points

Errors while devising the approach

Students often misinterpret the problem as asking to convert specific temperature values (like converting \(45°\mathrm{F}\) to Celsius). They might try to directly apply the formula \(9\mathrm{C} = 5(\mathrm{F} - 32)\) with \(\mathrm{F} = 45\), not realizing the question is about the difference between temperature extremes, not a specific temperature reading.

Some students might think this means one temperature was \(45°\mathrm{F}\), rather than understanding it means the gap between the highest and lowest temperatures was \(45°\mathrm{F}\). This leads to solving for a single temperature conversion rather than a temperature difference conversion.

Errors while executing the approach

When subtracting the two temperature equations [\(9\mathrm{C}_1 = 5(\mathrm{F}_1 - 32)\) and \(9\mathrm{C}_2 = 5(\mathrm{F}_2 - 32)\)], students often make mistakes in expanding and simplifying. Common errors include incorrectly handling the distribution of the subtraction or making sign errors when the constants (\(-160\) and \(+160\)) cancel out.

Even with the correct setup \(9 \times \text{(Celsius difference)} = 5 \times 45 = 225\), students may make simple division errors when calculating \(225 \div 9 = 25\). They might incorrectly compute this as 22.5 or make other arithmetic mistakes.

Errors while selecting the answer

Students might correctly calculate \(225 \div 9\) but write it as 25 (which is choice C). However, they might second-guess themselves and select choice A (\(65/9 \approx 7.22\)) thinking they need to express the answer as a fraction, not recognizing that 25 is the correct simplified form.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient temperature values

Let's say the temperature extremes on a particular day were:

• Low temperature: \(32°\mathrm{F}\) (this is convenient because \(32°\mathrm{F} = 0°\mathrm{C}\))
• High temperature: \(32 + 45 = 77°\mathrm{F}\)

The difference is \(77°\mathrm{F} - 32°\mathrm{F} = 45°\mathrm{F}\) ✓

Step 2: Convert both temperatures to Celsius

Using the given equation \(9\mathrm{C} = 5(\mathrm{F} - 32)\):

For the low temperature (\(32°\mathrm{F}\)):

\(9\mathrm{C} = 5(32 - 32) = 5(0) = 0\)

Therefore: \(\mathrm{C} = 0°\mathrm{C}\)

For the high temperature (\(77°\mathrm{F}\)):

\(9\mathrm{C} = 5(77 - 32) = 5(45) = 225\)

Therefore: \(\mathrm{C} = 225/9 = 25°\mathrm{C}\)

Step 3: Find the difference in Celsius

Temperature difference in Celsius = \(25°\mathrm{C} - 0°\mathrm{C} = 25°\mathrm{C}\)

Why this works: By choosing \(32°\mathrm{F}\) as our starting point, we made the conversion calculations clean and straightforward. The key insight is that regardless of which specific temperatures we choose (as long as they differ by \(45°\mathrm{F}\)), the difference in Celsius will always be the same due to the linear relationship between the scales.