Problem 64 - 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. $$77^{\circ} \mathrm{F}$$

Step-by-Step Solution

Answer

25°C

Step 1: Identify the Given Temperature

Identify the given temperature in Fahrenheit, which is 77°F.

Step 2: Use the Conversion Formula

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

Step 3: Substitute the Given Temperature

Substitute the given temperature into the formula: \[ C = \frac{5}{9}(77 - 32) \]

Step 4: Perform the Calculation Inside the Parentheses

Calculate the value inside the parentheses: \[ 77 - 32 = 45 \]

Step 5: Multiply and Divide

Multiply and divide to find the Celsius temperature: \[ C = \frac{5}{9} \times 45 \]\[ C = 25 \]

Step 6: Round the Answer If Necessary

Since the calculation results in an exact number, no rounding is necessary.

Key Concepts

headline of the respective core concept

When converting temperatures, knowing the formulas to switch between Celsius and Fahrenheit is important. First, identify the temperature you have. In this case, we start with 77°F.

Celsius to Fahrenheit conversion

To convert a temperature from Celsius to Fahrenheit, you use the formula: \[ F = \frac{9}{5}C + 32 \] The conversion is simple. Here’s a step-by-step process:

\[ Start with the temperature in Celsius \] Multiply the Celsius temperature by 9/5 (or 1.8) \[ Add 32 to the result to get the temperature in Fahrenheit \]

For example, let's convert 25°C to Fahrenheit:

\[ F = \frac{9}{5} \times 25 + 32 \] \[ F = 45 + 32 \] \[ F = 77^\text{°F} \]

Knowing this formula is useful for many scientific and everyday applications. It helps you compare weather forecasts in different countries and is essential for science experiments involving temperature.

Fahrenheit to Celsius conversion

For converting Fahrenheit to Celsius, use the formula: \[ C = \frac{5}{9}(F - 32) \] This formula is straightforward to use. Follow these steps:

\[ Start with the temperature in Fahrenheit \] Subtract 32 from the Fahrenheit temperature \[ Multiply the result by 5/9 \]

Take, for instance, converting 77°F to Celsius, as we did earlier:

\[ C = \frac{5}{9}(77 - 32) \] \[ 77 - 32 = 45 \] \[ C = \frac{5}{9} \times 45 \] \[ C = 25^{\text{°C}} \]

This conversion is helpful, especially when traveling between countries that use different temperature scales. Also, it is critical in fields such as cooking, science, and engineering, where accurate temperature measurements are crucial.

Metric system temperature

The metric system, widely used around the world, includes Celsius as the unit for temperature.
\[ Celsius is based on the freezing and boiling points of water. \]
\[ Water freezes at 0°C and boils at 100°C under standard atmospheric pressure. \] The Celsius scale is part of the International System of Units (SI), the modern form of the metric system. It is widely used in science and most countries. In contrast, the Fahrenheit scale, commonly used in the United States, defines water freezing at 32°F and boiling at 212°F. The conversion between these scales ensures that we can move between different systems of measurement easily. Understanding these conversions is important for global communication in science, travel, and trade. It allows us to better understand weather, environmental data, and various scientific contexts across the globe.