Problem 63 - 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. $$50^{\circ} \mathrm{F}$$

Step-by-Step Solution

Answer

10 °C

Step 1 - Understand the Conversion Formula

The given temperature needs to be converted from Fahrenheit to Celsius. The formula to convert Fahrenheit to Celsius is: \[ C = \frac{5}{9}(F-32) \]

Step 2 - Substitute the Given Value

Substitute the given Fahrenheit temperature (50 °F) into the formula: \[ C = \frac{5}{9}(50-32) \]

Step 3 - Calculate the Difference

First, calculate the difference between 50 and 32: \[ 50 - 32 = 18 \]

Step 4 - Multiply by the Fraction

Next, multiply the result by \( \frac{5}{9} \): \[ C = \frac{5}{9} \times 18 \]

Step 5 - Perform the Multiplication

Finally, calculate the multiplication: \[ C = 10 \]

Step 6 - Round the Answer

Since the result is a whole number, rounding is not required. The final answer is 10 °C.

Key Concepts

Celsius to Fahrenheit conversion

Understanding how to convert from Celsius to Fahrenheit is essential in temperature calculations. The formula is simple, but it helps to break it down. The conversion equation is: \[ F = \frac{9}{5} C + 32 \] Here’s how it works:
- Start with your temperature in Celsius (C).
- Multiply this by 9/5 (or 1.8).
- Add 32 to the result.
This formula reverses the process used to convert Fahrenheit to Celsius. Remember, always follow the order of operations: multiplication first, then addition.

Fahrenheit to Celsius conversion

Let’s take the specific example from our exercise. Converting a temperature from Fahrenheit to Celsius uses the formula: \[ C = \frac{5}{9} (F - 32) \] To break it down:
- Start with your Fahrenheit (F) temperature.
- Subtract 32 from the Fahrenheit value.
- Multiply the result by 5/9.
This will give you the temperature in Celsius. For example, converting 50°F into Celsius involves the following steps:
- 50 - 32 = 18
- multiplying 18 by \( \frac{5}{9} \) yields 10, so 50°F is equivalent to 10°C.

Metric system of measurements

The metric system is a standardized system of measurements used by most countries worldwide. It’s based on the International System of Units (SI).
Key benefits of the metric system include:

• Consistency: Follows decimal-based calculations for easier understanding.
• Universality: Widely recognized and used internationally.
• Simplicity: Units for measurements like length, mass, and temperature have logical intervals.

Temperature in the metric system is measured in degrees Celsius. This contrasts with the Fahrenheit scale used in the United States. Celsius is part of the broader effort to standardize measurements globally, making scientific communication clearer and more precise.

Temperature calculations

Accurate temperature calculations are crucial in science, cooking, and various real-life applications. Depending on the region and context, you might need to convert temperatures between Celsius and Fahrenheit.
To get these conversions right, remember these key points:

• Formulas: Use the correct conversion formulas. For Fahrenheit to Celsius, subtract 32 and multiply by 5/9. For Celsius to Fahrenheit, multiply by 9/5 and add 32.
• Rounding: Be mindful of rounding rules if required. Often, answers need to be rounded to the nearest tenth.
• Experience: Practice with different temperature values to become more comfortable with the conversions.

With these steps and a solid understanding of the formulas, temperature conversions become a manageable task.