Problem 47 - Elementary Algebra

Question

Convert the following temperatures to degrees Celsius given \(C=59(F-32),\) where F represents degrees Fahrenheit. $$ -13^{\circ} \mathrm{F} $$

Step-by-Step Solution

Answer

-13°F is -25°C.

Step 1: Identify the formula

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

Step 2: Substitute the given Fahrenheit temperature

The Fahrenheit temperature given is \(-13^{\circ} \mathrm{F}\). Substitute \( F = -13 \) into the formula to start the conversion process.

Step 3: Calculate the expression inside the parentheses

First, calculate \( F - 32 \). Substitute the value of \( F \): \( -13 - 32 = -45 \).

Step 4: Apply the multiplication by the fraction

Now, calculate the final Celsius temperature by using the formula \( C = \frac{5}{9} (-45) \). Multiply the fraction by \(-45\): \( \frac{5}{9} \times (-45) = \frac{-225}{9} \).

Step 5: Simplify the fraction

Simplify \( \frac{-225}{9} \) by dividing \(-225\) by \(9\): \( \frac{-225}{9} = -25 \). So the temperature in Celsius is \(-25^{\circ} \mathrm{C}\).

Key Concepts

Degrees Fahrenheit

Degrees Fahrenheit, often symbolized as °F, is a temperature scale used predominantly in the United States. It was developed by Daniel Gabriel Fahrenheit in the early 18th century. This scale sets 32°F as the freezing point of water and 212°F as its boiling point at standard atmospheric pressure. Fahrenheit is not commonly used worldwide, but it's important to understand, especially if you're dealing with international measurements.

• Useful for weather forecasts in specific regions.
• Based on intervals of 180 degrees between water's freezing and boiling points.
• Originally defined by setting human body temperature to approximately 96°F.

Knowing how to convert Fahrenheit to other scales like Celsius is crucial in global contexts.

Degrees Celsius

Degrees Celsius, noted as °C, is the most widely used temperature scale, adopted by most countries globally. It was conceived by Anders Celsius, a Swedish astronomer. This scale designates 0°C as the freezing point of water and 100°C as the boiling point, making it straightforward and intuitive.

• Easy to understand due to its simple 0-100 scale.
• Commonly used in scientific research and international travel.

Understanding Celsius is important for global communication and scientific consistency. It's often used alongside the metric system.

Conversion Formula

The conversion formula bridges the gap between Fahrenheit and Celsius. The formula is:\[ C = \frac{5}{9} (F - 32) \]This equation helps you convert any temperature from Fahrenheit (F) to Celsius (C). The formula reflects the difference in scale magnitudes, ensuring accurate conversions across temperature ranges.

• The subtraction of 32 aligns the scales' starting points (freezing water).
• The multiplier \( \frac{5}{9} \) adjusts for the different interval sizes between both scales.

Using this conversion formula is essential for applications in science, education, and daily life across different regions.

Algebraic Manipulation

Algebraic manipulation is a mathematical technique used to rearrange and simplify equations. It's key in converting temperatures using the formula between Fahrenheit and Celsius.
In our example, we begun with:\[ C = \frac{5}{9} (F - 32) \]By substituting the given Fahrenheit value (-13°F), you calculate:

• Subtract 32 from -13: \[ -13 - 32 = -45 \]
• Multiply the result by \( \frac{5}{9} \):\[ \frac{5}{9} imes (-45) = -25 \]

These steps illustrate how algebraic manipulation ensures precise calculation and highlights the importance of understanding each part of the process for accurate results.