Problem 41 - Algebra for College Students

Question

As of 2017 , the highest temperature ever recorded in the state of Washington was \(42^{\circ} \mathrm{C}\). Find the corresponding Fahrenheit temperature to the nearest degree.

Step-by-Step Solution

Answer

The corresponding Fahrenheit temperature is 108°F.

Step 1: Understand the Formula for Conversion

To convert from Celsius to Fahrenheit, use the formula \[ F = \frac{9}{5}C + 32 \] where F is the temperature in Fahrenheit and C is the temperature in Celsius.

Step 2: Substitute the Given Value

Substitute the given Celsius temperature into the conversion formula. In this case, the given temperature is 42°C. \[ F = \frac{9}{5} \times 42 + 32 \]

Step 3: Perform the Multiplication

First, multiply \(\frac{9}{5}\) by 42. \[ \frac{9}{5} \times 42 = \frac{378}{5} = 75.6 \]

Step 4: Add 32

Next, add 32 to the result from the previous step. \[ 75.6 + 32 = 107.6 \]

Step 5: Round to the Nearest Degree

Finally, round 107.6 to the nearest degree. \[ 107.6 \rightarrow 108 \]

Key Concepts

temperature conversion formulas

Converting temperatures between Celsius and Fahrenheit is a fundamental aspect of many scientific and everyday calculations. To do this, we use a formula that incorporates factors of conversion based on the scales' definitions. Understanding these formulas helps us accurately switch between these units.

The formula for converting Celsius to Fahrenheit is:
\( F = \frac{9}{5}C + 32 \)
where **F** represents the temperature in Fahrenheit and **C** represents the temperature in Celsius.

This formula operates in two main steps:

• First, multiply the temperature in Celsius by \( \frac{9}{5} \).
• Then, add 32 to this result.

This equation arises because the temperature scales have different zero points and increments. The 32 accounts for the difference in the zero points, while \( \frac{9}{5} \) adjusts for the differing increments.

Celsius to Fahrenheit calculation

Calculating the conversion from Celsius to Fahrenheit involves straightforward arithmetic. Let's use the highest temperature ever recorded in the state of Washington, which is 42°C, as an example.

1. **Substitute the Given Value**:
We start with the given temperature, which is 42°C, and substitute it into the formula:
\( F = \frac{9}{5} \times 42 + 32 \).

2. **Perform the Multiplication**:
We need to calculate \( \frac{9}{5} \) of 42 first. This step involves simple multiplication:
\( \frac{9}{5} \times 42 = 75.6 \).

3. **Add 32**:
To adjust for the zero point differences between the scales, we add 32 to our previous result:
\( 75.6 + 32 = 107.6 \).

4. **Round to the Nearest Degree**:
Finally, we round the result to the nearest degree to get the final Fahrenheit temperature:
107.6 rounds to 108.

Thus, 42°C is approximately 108°F.

rounding numbers in mathematical problems

Rounding is an essential skill in mathematics, which helps simplify numerical results. It's particularly useful in practical problems where overly precise numbers are unnecessary. To round a number to the nearest whole number, we follow a simple rule:

If the digit immediately after the decimal point is 5 or greater, we round up. If it is less than 5, we round down.

For instance, in our temperature conversion, we arrived at 107.6°F. The tenths place (6) is greater than 5, prompting us to round up to 108.

Rounding helps make results more readable and easier to communicate. It's used extensively in science, statistics, and everyday life. Consider when you split a bill among friends, the total amount is often rounded to simplify the payment process.

In summary:

• Identify the place to which you want to round.
• Look at the digit immediately to its right.
• Decide to round up or down based on that digit.

This process ensures that our results are both accurate and appropriately precise for the context.