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Chapter 2: Problem 47

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

Short Answer

Expert verified
-13掳F is -25掳C.

Step by step solution

01

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.
02

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.
03

Calculate the expression inside the parentheses

First, calculate \( F - 32 \). Substitute the value of \( F \): \( -13 - 32 = -45 \).
04

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 \(-45\): \( \frac{5}{9} \times (-45) = \frac{-225}{9} \).
05

Simplify the fraction

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees Fahrenheit
Degrees Fahrenheit, often symbolized as 掳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掳F as the freezing point of water and 212掳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掳F.
Knowing how to convert Fahrenheit to other scales like Celsius is crucial in global contexts.
Degrees Celsius
Degrees Celsius, noted as 掳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掳C as the freezing point of water and 100掳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掳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.

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