Problem 66 - Algebra and Trigonometry

Question

Temperature Scales The relationship between the Fahrenheit \((F)\) and Celsius \((C)\) temperature scales is given by the equation \(F=\frac{9}{5} C+32 .\) (a) Complete the table to compare the two scales at the given values. (b) Find the temperature at which the scales agree. [Hint: Suppose that \(a\) is the temperature at which the scales agree. Set \(F=a\) and \(C=a\) . Then solve for \(a . ]\)

Step-by-Step Solution

Answer

Complete the table using the equation; scales agree at \(-40^\circ\).

Step 1: Understanding the Equation

The given relationship between Fahrenheit and Celsius is described by the equation \(F=\frac{9}{5}C + 32\). This means that for any Celsius temperature \(C\), the corresponding Fahrenheit temperature \(F\) can be calculated by multiplying \(C\) by \(\frac{9}{5}\) and then adding 32.

Step 2: Completing the Table

To complete the table, you need to substitute the given Celsius values into the equation \(F=\frac{9}{5}C + 32\) to find the corresponding Fahrenheit values. For example:- If \(C = 0\), then \(F = \frac{9}{5}(0) + 32 = 32\).- If \(C = 20\), then \(F = \frac{9}{5}(20) + 32 = 36 + 32 = 68\).- If \(C = 100\), then \(F = \frac{9}{5}(100) + 32 = 180 + 32 = 212\).Complete the table with the Fahrenheit values for each given Celsius value.

Step 3: Solving for the Agreement Temperature

To find the temperature at which both scales agree, assume \(F = a\) and \(C = a\), meaning \(a = \frac{9}{5}a + 32\). We solve for \(a\) by first getting all terms involving \(a\) on one side: \[ a - \frac{9}{5}a = 32 \]This can be rewritten as:\[ \frac{5}{5}a - \frac{9}{5}a = 32 \]\[ -\frac{4}{5}a = 32 \]Now, solve for \(a\) by multiplying both sides by -5/4:\[ a = 32 \times \left(-\frac{5}{4}\right) = -40 \]

Step 4: Conclusion

The temperature at which both the Celsius and Fahrenheit scales show the same value is \(-40\) degrees. This means that at \(-40\) degrees, the Fahrenheit and Celsius temperatures are equal.

Key Concepts

Fahrenheit to Celsius conversion

Temperature conversion between Fahrenheit and Celsius is a useful skill, especially if you're traveling to a place that uses a different temperature scale. The basic formula to convert Celsius (C) to Fahrenheit (F) is given by:

• \[ F = \frac{9}{5}C + 32 \]

This formula means that whenever you know the temperature in Celsius, you can substitute it into the equation to find out the Fahrenheit equivalent. For example, if the temperature is 25°C, you can plug it into the equation to get:

• \[ F = \frac{9}{5} \times 25 + 32 = 77 \]

So, 25°C is equivalent to 77°F. Use this conversion formula any time you need to switch between temperature units.

Temperature equation

The fundamental formula that bridges the Celsius and Fahrenheit temperature scales is vital to understand. As stated above, the temperature equation is:

• \[ F = \frac{9}{5}C + 32 \]

This formula arises because the Fahrenheit and Celsius scales have different zero points and increments between degrees. By breaking down the components:

• \( \frac{9}{5} \) is the conversion factor to account for the different degree units.
• Adding 32 adjusts for the zero point difference, since 0°C equals 32°F.

Understanding this equation is key to performing accurate temperature conversions and grasping the relationship between these scales.

Equal temperature point

An intriguing concept is discovering the temperature at which Fahrenheit and Celsius readings intersect. This means they read the same number despite being different scales. To find this unique point, set Fahrenheit (F) equal to Celsius (C):

• Set \( F = a \) and \( C = a \)
• Solve using the equation \[ a = \frac{9}{5}a + 32 \]

By solving this equation, we find that both scales show the same value at \(-40\) degrees. This holds true mathematically and is a neat trivia fact for understanding temperature scales.

Solving equations

Solving the equation to find the equal point in Fahrenheit and Celsius involves understanding algebraic manipulation. Let's solve:

• Start from \[ a = \frac{9}{5}a + 32 \]
• Move terms involving \( a \) to one side: \[ a - \frac{9}{5}a = 32 \]
• Simplify: \[ \frac{5}{5}a - \frac{9}{5}a = 32 \]
• Resulting in \[ -\frac{4}{5}a = 32 \]
• Multiply both sides by \(-\frac{5}{4}\) to solve for \( a \): \[ a = -40 \]

This demonstrates equation-solving techniques such as combining like terms and isolating the variable, which are essential skills in mathematics.

Mathematical tables

Tabulating temperatures from Celsius to Fahrenheit can help reinforce comprehension. When provided with Celsius values, inserting them into the conversion formula allows for the creation of a mathematical table showing corresponding Fahrenheit values.Consider a table with the following Celsius values: 0, 20, and 100. Using the conversion equation \[ F = \frac{9}{5}C + 32 \], compute:

• For 0°C: \( F = 32 \)
• For 20°C: \( F = 68 \)
• For 100°C: \( F = 212 \)

Tables like these not only provide visual clarity but also offer an easy reference for common temperature conversions, which is especially helpful in scientific and daily applications.