Problem 5 - Discrete Mathematics with Applications

Question

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). $$g(0.27)$$

Step-by-Step Solution

Answer

Using the given formula \(F = \frac{9}{5}C + 32\), we substitute \(C = 0.27\) and calculate the Fahrenheit equivalent: \(F = \frac{9}{5}(0.27) + 32\). After the calculations, we find the result to be \(F = 32.486\), which can be rounded to \(F = 32.49\). Therefore, the temperature of \(0.27°C\) is approximately \(32.49°F\) on the Fahrenheit scale.

Step 1: Write down the given information.

We are given the formula to convert Celsius to Fahrenheit: \(F = \frac{9}{5}C + 32\) The Celsius temperature we need to convert is: \(C = 0.27\)

Step 2: Substitute the Celsius temperature value into the formula.

Using the given formula, we will replace \(C\) with \(0.27\): \(F = \frac{9}{5}(0.27) + 32\)

Step 3: Calculate the conversion.

Now, we will perform the calculations: \(F = \frac{9}{5}(0.27) + 32\) \(F = \frac{9 (0.27)}{5} + 32\) \(F = \frac{2.43}{5} + 32\) \(F = 0.486 + 32\) \(F = 32.486\)

Step 4: Round the result.

The number we got, 32.486, may be rounded to the nearest hundredth to give us a more precise result: \(F = 32.49\) The result is that the temperature of \(0.27°C\) is equivalent to approximately \(32.49°F\) on the Fahrenheit scale.

Key Concepts

Celsius to Fahrenheit Conversion

Temperature can be measured using different scales, and two of the most commonly known are Celsius and Fahrenheit. Understanding how to convert between them is critical, especially when encountering temperatures expressed in a less familiar scale.
To convert Celsius (°C) to Fahrenheit (°F), the formula used is:

• \( F = \frac{9}{5}C + 32 \)

This mathematical formula changes a Celsius value into its Fahrenheit equivalent.
Let's break it down:

• \( \frac{9}{5} \) is the conversion factor that scales the Celsius temperature to Fahrenheit.
• "+ 32" recalibrates the value because the two scales don't start at the same point. Interestingly, 0°C is 32°F, hence the +32 component.

For example, when converting 0.27°C:

• Multiply 0.27 by \( \frac{9}{5} \) to adjust the scale.
• Add 32 to shift the value to the correct Fahrenheit level.

Following these steps accurately will help you convert with ease.

Rounding Numbers

Mathematics often deals with numbers that have a lot of decimal places, but everyday use typically doesn't require excessive precision.
Rounding simplifies these numbers, making calculations and understandings more practical in real life.

• When you round a number, you can round it to the nearest whole number, tenth, hundredth, etc.

To round a number like 32.486 to the nearest hundredth:

• Identify the third digit after the decimal point – in this case, it's "6."
• If this digit is 5 or more, the second digit after the decimal point increases by one.
• So, 32.486 becomes 32.49, since "6" encourages rounding up the "8" to "9."

Remember, rounding ensures your calculations remain both manageable and adequately precise for most purposes.

Mathematical Formulas

Formulas are essential in mathematics and science, acting as the language that conveys relationships between different quantities.
They are typically expressed as equations.

• In our context, the formula \( F = \frac{9}{5}C + 32 \) shows the linear relationship between Celsius and Fahrenheit temperatures.

Understanding how to apply a formula involves:

• Identifying the variables (C for Celsius and F for Fahrenheit in this case).
• Substituting known values into the formula, as we did by inserting 0.27 for C.
• Performing any arithmetic operations required to solve for the unknown.

Using formulas effectively helps solve real-world problems and make sense of numerical data. In temperature conversion, they streamline the process, making it nearly automatic once you grasp their workings.