Problem 42 - Algebra for College Students

Question

As of \(2017,\) the lowest temperature ever recorded in South Dakota was \(-41^{\circ} \mathrm{F}\). Find the corresponding Celsius temperature to the nearest degree.

Step-by-Step Solution

Answer

The corresponding Celsius temperature is \( -41^{\circ} \text{C} \).

Step 1: Understand the Temperature Conversion Formula

The formula to convert Fahrenheit to Celsius is \( C = \frac{5}{9}(F - 32) \), where \( C \) is the temperature in Celsius and \( F \) is the temperature in Fahrenheit.

Step 2: Substitute the Given Fahrenheit Temperature into the Formula

Use the given temperature \( -41^{\circ} \mathrm{F} \) and substitute it into the formula: \[ C = \frac{5}{9}(-41 - 32) \]

Step 3: Perform the Subtraction Inside the Parenthesis

Calculate the expression inside the parenthesis: \( -41 - 32 = -73 \). Now the formula looks like: \[ C = \frac{5}{9}(-73) \]

Step 4: Multiply by the Fraction

Multiply \( -73 \) by the fraction \( \frac{5}{9} \): \[ C = \frac{5}{9} \times (-73) = \frac{-365}{9} \]

Step 5: Simplify the Fraction

Divide \( -365 \) by \( 9 \) to get a decimal: \[ -365 \div 9 \approx -40.56 \]

Step 6: Round to the Nearest Degree

Round \( -40.56 \) to the nearest degree, which is \( -41^{\circ} \text{C} \).

Key Concepts

Fahrenheit to Celsius conversion

When converting from Fahrenheit to Celsius, we use a specific formula. This formula helps us understand the relationship between these two temperature scales. The conversion formula is:
\[ C = \frac{5}{9}(F - 32) \]
Here, **C** stands for the temperature in Celsius, and **F** stands for the temperature in Fahrenheit. This formula works because it takes into account that the two scales have different zero points and different sizes of degrees. To convert, you first subtract 32 from the Fahrenheit temperature, then multiply the result by \( \frac{5}{9} \). This step-by-step transformation allows us to move from one measurement system to another seamlessly. Make sure to follow the order of operations correctly to get the accurate conversion result.

Temperature conversion formula

The temperature conversion formula is essential in scientific calculations and daily life to convert temperature readings from Fahrenheit to Celsius.
### Step-by-Step Calculation To use the formula correctly, follow these steps:

• Identify the temperature in Fahrenheit you want to convert. Let's say it's \( -41^{\circ} \text{F} \).


• Subtract 32 from the Fahrenheit temperature: \( -41 - 32 = -73 \).


• Multiply the result by \( \frac {5}{9} \). For our example, \( \frac{5}{9} \times -73 = \frac{-365}{9} \).


• Carry out the division to get the Celsius temperature: \( \frac{-365}{9} = -40.56 \).


• Round to the nearest degree if necessary: \( -40.56 \approx -41^{\circ} \text{C} \).

### Why the Formula Works The formula works because it bridges the gap between two different scales by standardizing the point at which water freezes (32°F) and then adjusting for the different increments in degrees.

Negative temperature conversion

Converting negative temperatures might seem tricky at first, but it follows the same process as positive temperatures. Let's illustrate with a real-life example.
### Example Calculation: Consider converting \(-41^{\circ} \text{F} \) to Celsius.

• Start with \( -41^{\circ} \text{F} \).
• Subtract 32: \( -41 - 32 = -73 \).
• Multiply by \( \frac{5}{9} \): \( -73 \times \frac{5}{9} = -40.56 \).
• Round to the nearest degree: \( -40.56 \approx -41^{\circ} \text{C} \).

Negative conversion follows the same steps: subtract 32, multiply by \( \frac{5}{9} \), and round. This consistency makes temperature conversion easier to remember and apply.
}]