Problem 62 - 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. $$200^{\circ} \mathrm{C}$$

Step-by-Step Solution

Answer

200^{\textdegree C} = 392^{\textdegree F}

Step 1: Apply the Conversion Formula

\(F = \frac{9}{5}C + 32\)

Step 2: Substitute

\(F = \frac{9}{5}(200) + 32 = 360 + 32 = 392\)

Step 3: State the Answer

\(200°C = 392°F\)

Key Concepts

Celsius to Fahrenheit Conversion

Converting temperatures from Celsius to Fahrenheit is a common task in science and everyday life. The formula to switch from Celsius to Fahrenheit is straightforward: \( F = \frac{9}{5} C + 32 \).
This means you take your Celsius temperature, multiply it by 9/5, and then add 32 to convert it to Fahrenheit.

Let's break it down with an example:
Suppose you have a temperature of 200°C and you need to find out how much that is in Fahrenheit:
\( F = \frac{9}{5} \times 200 + 32 \)
First, multiply 200 by 9/5, which equals 360.
Next, add 32 to 360.
So, \( F = 360 + 32 = 392 \).
Hence, 200°C is equal to 392°F. Remember to always follow the order of operations: first the multiplication, then the addition.

Fahrenheit to Celsius Conversion

Converting from Fahrenheit to Celsius is like flipping the previous formula. The conversion formula here is \( C = \frac{5}{9}(F - 32) \).
It starts with subtracting 32 from the Fahrenheit temperature and then multiplying by 5/9.

Let's use an example to clarify:
Suppose you have a temperature of 392°F and need to find out how much it is in Celsius:
\( C = \frac{5}{9}(392 - 32) \)
First, subtract 32 from 392, which equals 360.
Then, multiply 360 by 5/9.
So, \( C = \frac{5}{9} \times 360 = 200 \).
Therefore, 392°F is equal to 200°C. Conversion in this manner requires careful attention to the order: subtraction first, then multiplication.

Metric System

The Metric System is a decimal-based system of measurement used around the world, particularly in scientific contexts. It includes units like meters for distance, grams for weight, and degrees Celsius for temperature.

Advantages of the metric system include:

• Simplicity: Conversions within the metric system are seamless because they are all base-10.
• Universality: It's used globally, making scientific communication clearer.
• Consistency: Units are standardized, reducing confusion in measurements.

For temperature, the Celsius scale is used in the metric system. Water freezes at 0°C and boils at 100°C.
In contrast, the Fahrenheit scale, which is part of the Imperial system, has water freezing at 32°F and boiling at 212°F.
Understanding how to convert between Celsius and Fahrenheit thus becomes essential for working across different measurement systems.