Problem 45 - Elementary Algebra

Question

Convert the following temperatures to degrees Celsius given \(C=59(F-32),\) where F represents degrees Fahrenheit. $$ 86^{\circ} \mathrm{F} $$

Step-by-Step Solution

Answer

86°F is 30°C.

Step 1: Identify the Formula

The formula for converting Fahrenheit to Celsius is given as \( C = \frac{5}{9}(F-32) \). Here, \( F \) is the temperature in Fahrenheit which you want to convert to Celsius.

Step 2: Plug in the Fahrenheit Value

Substitute \( F = 86 \) into the formula. This gives: \[ C = \frac{5}{9}(86-32) \].

Step 3: Perform the Subtraction

Inside the parentheses, subtract 32 from 86: \( 86 - 32 = 54 \).

Step 4: Multiply by 5

Multiply the result by 5: \( 5 \times 54 = 270 \).

Step 5: Divide by 9

Now divide the result by 9: \( \frac{270}{9} = 30 \).

Step 6: Conclusion

Therefore, \( 86^{\circ} \text{F} \) is equivalent to \( 30^{\circ} \text{C} \).

Key Concepts

Temperature Conversion Formula

Converting temperatures from Fahrenheit to Celsius is essential in many scientific and everyday contexts. The formula that helps achieve this is quite straightforward but important: \[ C = \frac{5}{9}(F - 32) \]This equation is key when you want to convert a temperature measured in Fahrenheit (\( F \)) to Celsius (\( C \)).
Here's a quick breakdown of the components:

• The \( F \) stands for the Fahrenheit temperature that you're converting.
• The expression \((F - 32)\) subtracts 32 from the Fahrenheit temperature, adjusting it according to the formula.
• The fraction \( \frac{5}{9} \) adjusts the scale from Fahrenheit to Celsius.

Using this formula is simple. Plug in the Fahrenheit value, perform the arithmetic operations, and you have your temperature in Celsius.

Celsius Calculation

Let's focus on how you calculate temperatures in Celsius once you know the Fahrenheit value. Given a temperature in Fahrenheit, the first step in conversion is an inside arithmetic operation: subtraction.
Substitute the Fahrenheit value into the formula, like so: if \( F = 86 \), then you have:\[ C = \frac{5}{9}(86 - 32) \]Then calculate what’s inside the parenthesis:

• Subtract 32 from 86 to get 54.
• This gives you \( 54 \) left to work with.

This initial subtraction adjusts for the difference between the two temperature scales.
Following this, multiply by \( 5 \), and then divide the result by \( 9 \) to finish off the conversion. After these steps, the final Celsius calculation will give you the temperature in the needed metric unit.

Subtraction and Arithmetic Operations

Understanding subtraction and other basic arithmetic operations is vital for correctly using the temperature conversion formula.
1. Subtraction: This operation is first applied in the formula to adjust the Fahrenheit value:

• In the example, \( 86 - 32 = 54 \).

This step corrects the starting point by removing the baseline difference between the Fahrenheit and Celsius scales.
2. Multiplication: Next, the result is multiplied by \( 5 \):

• For example, \( 54 \times 5 = 270 \).

Multiplication scales the adjusted temperature back to Celsius size proportion but needs further refinement.
3. Division: Finally, the result from multiplication is divided by \( 9 \), which adjusts the temperature to the precise Celsius scale:

• \( \frac{270}{9} = 30 \)

These operations together make the conversion possible. Each arithmetic operation in combination with the previous brings you closer to the exact Celsius temperature.