Problem 65 - Precalculus Student Solutions Manual 5th

Question

Celsius and Fahrenheit Temperatures In the met- ric system of weights and measures, temperature is measured in degrees Celsius (" \(^{\circ}\) C) instead of degrees Fahrenheit \(\left(^{\circ} \mathrm{F}\right) .\) To convert between the two systems, we use the equations $$ C=\frac{5}{9}(F-32) \quad \text { and } \quad F=\frac{9}{5} C+32 $$ In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary. $$100^{\circ} \mathrm{F}$$

Step-by-Step Solution

Answer

100∘F equals 37.8∘C.

Step 1 - Identify the given temperature

The given temperature is in Fahrenheit, which is \(100^{\text{∘}}F\).

Step 2 - Use the Celsius conversion formula

Use the formula to convert Fahrenheit to Celsius: \ \[ C = \frac{5}{9} (F - 32) \]

Step 3 - Substitute the Fahrenheit value into the formula

Substitute \(100^{\text{∘}}F\) into the formula: \ \[ C = \frac{5}{9} (100 - 32) \]\(100 - 32 = 68\)

Step 4 - Simplify the equation

Calculate the resulting value: \ \[ C = \frac{5}{9} \times 68 \]\( \frac{5}{9} \times 68 \approx 37.78 \)

Step 5 - Round the answer

Round the answer to the nearest tenth: \ \(37.78\) rounded to the nearest tenth is \(37.8\) degrees Celsius.

Key Concepts

Celsius to Fahrenheit

Converting temperatures from Celsius to Fahrenheit is a common task, especially when dealing with different temperature measurement systems. In countries using the metric system, Celsius is standard, while Fahrenheit is used in the United States. The formula to convert Celsius to Fahrenheit is straightforward: \[ F = \frac{9}{5} C + 32 \] where \( C \) represents the temperature in degrees Celsius and \( F \) represents the temperature in degrees Fahrenheit. This formula requires you to first multiply the Celsius temperature by \( \frac{9}{5} \), a conversion factor, and then add 32 to this product.

• Example: Convert \( 25^{\text{∘}}C \) to Fahrenheit.
• First, multiply \( 25 \) by \( \frac{9}{5} \): \( 25 \times \frac{9}{5} = 45 \).
• Next, add 32 to the result: \( 45 + 32 = 77 \).
• Therefore, \( 25^{\text{∘}}C \) equals \( 77^{\text{∘}}F \).

As you can see, the conversion is simple once you understand the formula and the steps involved.

Fahrenheit to Celsius

Conversely, converting temperatures from Fahrenheit to Celsius is equally important and uses a slightly different formula. This is often necessary when working in scientific or international contexts. The conversion formula is: \[ C = \frac{5}{9}(F - 32) \] where \( F \) is the temperature in degrees Fahrenheit and \( C \) is the temperature in degrees Celsius. Here's a step-by-step guide:

• Start with the Fahrenheit temperature you want to convert.
• Subtract 32 from the Fahrenheit value.
• Multiply the result by \( \frac{5}{9} \).
• The resulting product is the temperature in degrees Celsius.

For example, let’s convert \( 68^{\text{∘}}F \) to Celsius:

• First, subtract 32 from 68: \( 68 - 32 = 36 \).
• Then multiply by \( \frac{5}{9} \): \( 36 \times \frac{5}{9} \approx 20 \).
• So, \( 68^{\text{┰}}F \) is equivalent to \( 20^{\text{┰}}C \).

This formula helps in making quick and accurate conversions from Fahrenheit to Celsius.

Metric System

The metric system is an international system of measurement that is used in nearly every country except the United States. It is simple and based on powers of ten, making conversions straightforward and intuitive. Temperature measurement in the metric system is done in degrees Celsius ( ∘C), as opposed to the degrees Fahrenheit ( ∘F) used in the imperial system.

• The metric system is uniform and facilitates communication and trade between countries.
• It includes units like the meter for length, kilogram for mass, and liter for volume.
• Scientific work almost exclusively uses the metric system for precision and ease of conversion.

To understand why different systems are used, consider historical contexts. The metric system was designed during the French Revolution to unify different measures in France. In contrast, the Fahrenheit scale was created by Daniel Gabriel Fahrenheit based on average human body temperature. Adopting the metric system for temperature, as with other measurements, ensures clarity and ease whether in academic scenarios or everyday use.