Problem 52 - Calculus for Biology and Medicine

Question

To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. We discuss these scales in Problems 52 and \(53 .\) (a) The Celsius scale is devised so that \(0^{\circ} \mathrm{C}\) is the freezing point of water (at 1 atmosphere of pressure) and \(100^{\circ} \mathrm{C}\) is the boiling point of water (at 1 atmosphere of pressure). If you are more familiar with the Fahrenheit scale, then you know that water freezes at \(32^{\circ} \mathrm{F}\) and boils at \(212^{\circ} \mathrm{F}\). Find a linear equation that relates temperature measured in degrees Celsius and temperature measured in degrees Fahrenheit. (b) The normal body temperature in humans ranges from \(97.6^{\circ} \mathrm{F}\) to \(99.6^{\circ} \mathrm{F}\). Convert this temperature range into degrees Celsius.

Step-by-Step Solution

Answer

The Celsius-Fahrenheit equation is \(F = 1.8C + 32\). The body temperature range in Celsius is 36.4°C to 37.6°C.

Step 1: Understand the problem

There are two temperature scales to be compared: Celsius and Fahrenheit. We need to find a linear equation that relates these two scales and use it to convert body temperature range from Fahrenheit to Celsius.

Step 2: Set points for linear relation

Identify the corresponding points for each scale: on Celsius, - Freezing point: \(0^{\circ} C\) at \(32^{\circ} F\)- Boiling point: \(100^{\circ} C\) at \(212^{\circ} F\).

Step 3: Establish the linear equation

Considering a linear relationship of form \(F = mC + b\):1. Using boiling point: \(212 = m(100) + b\)2. Using freezing point: \(32 = m(0) + b\).From the boiling point equation, substitute \(b\) from freezing point equation.

Step 4: Solving for slope (m)

The slope \(m\) can be calculated using the two points:\(m = \frac{(212 - 32)}{(100 - 0)} = \frac{180}{100} = 1.8\).

Step 5: Solve for intercept (b)

Substitute \(m = 1.8\) into \(32 = 1.8(0) + b\), so \(b = 32\).Thus, the equation is \(F = 1.8C + 32\).

Step 6: Rearrange for Celsius

Rearrange the equation for conversion from Fahrenheit to Celsius:\(C = \frac{(F - 32)}{1.8}\).

Step 7: Convert temperature range

Convert 97.6°F and 99.6°F to Celsius:1. For 97.6°F: \(C = \frac{(97.6 - 32)}{1.8} \approx 36.4^{\circ} C\).2. For 99.6°F: \(C = \frac{(99.6 - 32)}{1.8} \approx 37.6^{\circ} C\).

Step 8: Conclusion

The temperature range in Celsius is approximately 36.4°C to 37.6°C.

Key Concepts

Celsius and Fahrenheit Relationship

Temperature is a daily term and understanding how Celsius and Fahrenheit relate can make scientific discussions a lot easier. The Celsius and Fahrenheit scales both define the temperature at which water freezes and boils, but they do so at different values. In Celsius, water freezes at 0°C and boils at 100°C. In Fahrenheit, these points are set at 32°F for freezing and 212°F for boiling.

To convert between these two scales, we use a linear equation derived from their fixed points. The formula to convert Celsius (C) to Fahrenheit (F) is:

1. Start with the Celsius temperature.
2. Multiply by 1.8.
3. Add 32 to the result.

This linear equation is written as: \[F = 1.8C + 32\]
Inverting this, for converting Fahrenheit to Celsius, is:\[C = \frac{(F - 32)}{1.8}\] These equations show a direct relationship, meaning small changes in Celsius are reflected proportionately in Fahrenheit, based on the slope and intercept line established between the two points.

Linear Equations in Temperature

Linear equations offer a simple way to understand relationships between variables. When it comes to temperature, the relationship between Celsius and Fahrenheit can be closely represented by a linear equation. Why is this the case?

Let's break it down:

• The freezing and boiling points establish two key reference points.
• These points show that for every degree Celsius, the Fahrenheit temperature changes by 1.8 degrees.
• The slope (1.8) reflects how steep the change from Celsius is to Fahrenheit.

The form \( F = mC + b \) fits this perfectly, where \( m \) is the slope (1.8) and \( b \) is the y-intercept (32). These constants effectively capture the rate of change and starting point of Fahrenheit temperature with respect to Celsius.

In summary, understanding these linearly related equations helps in conversion and provides insight into the physics of temperature translation because both temperature scales follow a consistent, predictable pattern.

Temperature Scales Comparison

Delving deeper into temperature scales reveals that Celsius and Fahrenheit are two of the most commonly used scales worldwide. But how do these two compare with an additional scale called Kelvin, frequently used in scientific contexts?

• Celsius (°C): Designed around the freezing (0°C) and boiling (100°C) points of water, Celsius is ideal for everyday weather and simple scientific observations.
• Fahrenheit (°F): Most often used in the United States for weather-related purposes, Fahrenheit is based on a broader scale where water freezes at 32°F and boils at 212°F.
• Kelvin (K): The Kelvin scale starts at absolute zero, the point at which all molecular motion stops. This makes it invaluable for scientific calculations. It is linearly related to Celsius by the equation \( K = C + 273.15 \).

Comparing these scales, each one serves unique purposes. While Celsius and Fahrenheit offer everyday practicality, Kelvin provides crucial scientific utility.

In essence, recognizing how each scale measures temperature can aid in selecting the appropriate context and making accurate conversions in scientific research or daily tasks.