Problem 3 - University Physics with Modern Physics

Question

While vacationing in Italy, you see on local TV one summer morning that temperature will rise from the current \(18^{\circ} \mathrm{C}\) to a high of \(39^{\circ} \mathrm{C}\) . What is the corresponding increase in the Fahrenheit temperature?

Step-by-Step Solution

Answer

The increase in Fahrenheit temperature is \(37.8^{\circ} \mathrm{F}\).

Step 1: Understand the Celsius to Fahrenheit Formula

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

Step 2: Convert Initial Celsius Temperature to Fahrenheit

First, convert the initial temperature of \(18^{\circ} \mathrm{C}\) to Fahrenheit:\[ F_1 = \frac{9}{5} \times 18 + 32 \]Calculating gives:\[ F_1 = 32.4 + 32 = 64.4^{\circ} \mathrm{F} \]

Step 3: Convert Final Celsius Temperature to Fahrenheit

Next, convert the high temperature of \(39^{\circ} \mathrm{C}\) to Fahrenheit:\[ F_2 = \frac{9}{5} \times 39 + 32 \]Calculating gives:\[ F_2 = 70.2 + 32 = 102.2^{\circ} \mathrm{F} \]

Step 4: Calculate the Increase in Fahrenheit

Subtract the initial Fahrenheit temperature from the final Fahrenheit temperature:\[ \Delta F = F_2 - F_1 = 102.2 - 64.4 \]This calculation results in:\[ \Delta F = 37.8^{\circ} \mathrm{F} \]

Key Concepts

Celsius to Fahrenheit Formula

When you want to convert a temperature from Celsius to Fahrenheit, there's a straightforward formula that you can use: \( F = \frac{9}{5}C + 32 \). This means you'll need to multiply the Celsius temperature by \( \frac{9}{5} \) and then add 32 to the result.
For example, if you have a temperature of \(10^{\circ} \mathrm{C}\), you would calculate \( F = \frac{9}{5} \times 10 + 32 = 18 + 32 = 50^{\circ} \mathrm{F} \).
It's a good idea to memorize this formula for converting Celsius to Fahrenheit, as it can be very handy, especially when traveling or dealing with weather forecasts in different regions.

Fahrenheit Temperature Calculation

Converting temperatures from Celsius to Fahrenheit involves calculating each temperature separately using the formula mentioned earlier.
Let's look at the given example: You start with \(18^{\circ} \mathrm{C}\), converting to Fahrenheit using \( F_1 = \frac{9}{5} \times 18 + 32 \).
Breaking it down:

• Multiply 18 by \(\frac{9}{5} \) to get 32.4.
• Add 32 to 32.4 to get 64.4. So, \(18^{\circ} \mathrm{C}\) is the same as \(64.4^{\circ} \mathrm{F}\).

Next, for \(39^{\circ} \mathrm{C}\), use the formula: \( F_2 = \frac{9}{5} \times 39 + 32 \).
Breaking down this calculation:

• Multiply 39 by \(\frac{9}{5} \) to get 70.2.
• Add 32 to 70.2 to reach 102.2. Thus, \(39^{\circ} \mathrm{C}\) is \(102.2^{\circ} \mathrm{F}\).

These simple steps help convert temperatures accurately for any situation.

Temperature Increase Calculation

Once you've converted both Celsius temperatures into Fahrenheit, calculating the increase in temperature becomes easy.
You simply subtract the initial Fahrenheit temperature from the final Fahrenheit temperature. In the exercise, you started with \(64.4^{\circ} \mathrm{F}\) and concluded with \(102.2^{\circ} \mathrm{F}\).
To find the increase, compute:

• \(\Delta F = F_2 - F_1 = 102.2 - 64.4 = 37.8^{\circ} \mathrm{F}\).

This calculation reveals that the Fahrenheit temperature will increase by \(37.8^{\circ} \mathrm{F}\).
Understanding this process is essential for grasping how temperature changes are measured across different temperature scales.