Problem 52 - Calculus for Biology and Medicine

Question

(a) To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. The Celsius scale is devised so that \(0^{\circ} \mathrm{C}\) is the freezing point of water and \(100^{\circ} \mathrm{C}\) is the boiling point of water. 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. (c) Is there any temperature that reads the same in Celsius and Fahrenheit?

Step-by-Step Solution

Answer

(a) The equation is \( F = 1.8C + 32 \). (b) The range is 36.44°C to 37.56°C. (c) -40 degrees is the same on both scales.

Step 1: Understanding the Temperature Points

To find the relationship between Celsius (C) and Fahrenheit (F), we first note that the freezing point of water is at 0°C (32°F) and the boiling point of water is at 100°C (212°F). These points will help us form the linear equation between the scales.

Step 2: Find the Slope of the Relationship

Using the points (0, 32) and (100, 212), calculate the slope (m) of the line:\( m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \).This slope means that for every 1 degree increase in Celsius, the temperature in Fahrenheit increases by 1.8 degrees.

Step 3: Forming the Linear Equation

With a slope of 1.8 and the point (0, 32), we use the equation of a line, \( F = m \cdot C + b \). Insert the known values: \( 32 = 1.8 \cdot 0 + b \), solving gives \( b = 32 \). Thus, the linear equation is \( F = 1.8C + 32 \).

Step 4: Convert Normal Body Temperature Range to Celsius

Using the equation \( F = 1.8C + 32 \), rearrange it to find C: \( C = \frac{F - 32}{1.8} \).Convert each limit of the range:- Lower: \( C = \frac{97.6 - 32}{1.8} \approx 36.44^\textrm{C}\).- Upper: \( C = \frac{99.6 - 32}{1.8} \approx 37.56^\textrm{C}\). Thus, the range in Celsius is \( 36.44^\textrm{C} \) to \( 37.56^\textrm{C} \).

Step 5: Finding the Same Temperature in Both Scales

Set the temperatures equal: \( F = C \), and use the equation \( F = 1.8C + 32 \).Set \( C = F \): \( C = 1.8C + 32 \).Rearranging gives \( 0.8C = -32 \).Solving gives \( C = \frac{-32}{0.8} = -40 \).Thus, both scales read -40 degrees at the same temperature.

Key Concepts

Understanding Linear Equations

A linear equation is a mathematical statement that shows the relationship between two variables, forming a straight line when graphed. In the context of temperature conversion, this relationship is depicted between Celsius (C) and Fahrenheit (F). The equation typically takes the form of a simple line equation, expressed as:

• \( y = mx + b \)

where \( m \) is the slope and \( b \) is the y-intercept.
The slope represents how much one unit change in x affects y. In this case, it shows how degrees in Celsius correspond to degrees in Fahrenheit.
To calculate the slope \( m \) in any linear equation, use the formula:

• \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)

In our example, the points of interest are the freezing and boiling points of water on the Celsius and Fahrenheit scales: (0, 32) and (100, 212). By substituting these points, we find the slope to be 1.8. This indicates a direct proportion where a rise in temperature by 1°C corresponds to an increase by 1.8°F. Understanding this crucial point stresses the importance of linear equations in converting temperature accurately.

Converting Celsius to Fahrenheit

The Celsius and Fahrenheit temperature scales are widely used across different regions of the world.
To convert a temperature from Celsius to Fahrenheit, we use a straightforward linear equation derived from known reference points, which is:

• \( F = 1.8C + 32 \)

This formula provides a practical way to switch temperatures by multiplying the Celsius value by 1.8 and then adding 32.
For example, to convert 25°C to Fahrenheit, you calculate:

• \( F = 1.8 \times 25 + 32 = 77°F \)

Understanding this equation prepares you to seamlessly convert between the scales in scientific calculations or geographical data discussions.

Converting Fahrenheit to Celsius

When working with temperatures measured in Fahrenheit, converting them to Celsius involves rearranging the linear equation:

• \( C = \frac{{F - 32}}{1.8} \)

This formula helps simplify the conversion process. Simply subtract 32 from the Fahrenheit temperature, then divide the result by 1.8 to find the corresponding degrees Celsius.
Let’s take an example of converting 77°F to Celsius. By using our formula, the conversion would be:

• \( C = \frac{{77 - 32}}{1.8} \approx 25°C \)

Moreover, a fascinating fact is that -40 is the unique temperature reading identically in both Celsius and Fahrenheit scales. Using both conversion equations can help confirm this remarkable coincidence.
The simplicity and logic behind these conversion formulas make them essential tools in today’s global interactions, especially in fields requiring precise temperature data.