Problem 61 - 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. $$40^{\circ} \mathrm{C}$$

Step-by-Step Solution

Answer

40\textdegree C = 104\textdegree F

Step 1 - Identify the conversion formula

To convert from degrees Celsius (\textdegree C) to degrees Fahrenheit (\textdegree F), use the formula: \[ F = \frac{9}{5}C + 32 \]

Step 2 - Plug in the given Celsius value

Substitute the given temperature (40\textdegree C) into the formula: \[ F = \frac{9}{5}(40) + 32 \]

Step 3 - Perform the multiplication

First, calculate the multiplication: \[ \frac{9}{5} \times 40 = 72 \]

Step 4 - Add 32 to the result

Now, add 32 to the product from the previous step: \[ 72 + 32 = 104 \]Thus, 40\textdegree C is equivalent to 104\textdegree F.

Key Concepts

Celsius to Fahrenheit

Temperature conversion between Celsius and Fahrenheit is a fundamental concept in the study of thermodynamics and weather patterns. Celsius, used predominantly in most countries around the world, is based on the metric system. Fahrenheit, still used in the United States, follows the imperial system. To convert from Celsius to Fahrenheit, you can use this simple formula:

• First, identify the conversion formula: \( F = \frac{9}{5}C + 32 \).

By substituting the Celsius temperature (e.g., 40°C) into the formula, you can follow these steps:

• Plug in the given Celsius value: \( F = \frac{9}{5}(40) + 32 \).
• Perform the multiplication: \( \frac{9}{5} \times 40 = 72 \).
• Add 32 to the result: \( 72 + 32 = 104 \).

Therefore, 40°C is equivalent to 104°F.

This conversion can be very helpful in many real-life situations, such as preparing for travel to countries using different temperature units or cooking using international recipes.

Temperature Formulas

Understanding temperature formulas is essential for converting between Celsius and Fahrenheit. The conversion formulas are:

• From Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \).
• From Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \).

The formulas are designed to accommodate the differences between the two scales.
The key steps for these conversions involve:

• Identifying the correct formula based on the given temperature.
• Substituting the given temperature into the formula.
• Performing the necessary arithmetic operations.

For example, converting 40°C to Fahrenheit involves multiplying by 9/5 and then adding 32. Conversely, converting 104°F to Celsius involves subtracting 32 and then multiplying by 5/9.

Being proficient with these formulas allows you to understand and compare temperatures effectively in different units.

Metric and Imperial Systems

The world utilizes two primary systems of measurement: the metric system and the imperial system. The metric system, more commonly referred to as the International System of Units (SI), is used by most countries globally. It includes measurements like meters for length, kilograms for mass, and Celsius for temperature.

Conversely, the imperial system is mainly used in the United States. This system includes inches for length, pounds for mass, and Fahrenheit for temperature.

Understanding both systems allows you to interpret and convert measurements more accurately. Temperature conversion between Celsius (metric) and Fahrenheit (imperial) is particularly crucial for:

• Weather reports.
• Scientific research.
• Cooking and baking recipes.

By comprehending the basic conversion formulas, one can easily switch between these two systems, making international communication smoother and more precise.