Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

The relationship between the temperature reading \(F\) on the Fahrenheit scale and the temperature reading \(C\) on the Celsius scale is given by \(C=\frac{5}{9}(F-32)\). (a) Find the temperature at which the reading is the same on both scales. (b) When is the Fahrenheit reading twice the Celsius reading?

Step-by-Step Solution

Answer

(a) -40°C/F; (b) 320°F, 160°C.

Step 1: Set the Temperature Readings Equal

To find the temperature at which the readings are the same on both the Celsius and Fahrenheit scales, set the equation \( C = \frac{5}{9}(F-32) \) equal to \( F \). This gives us the equation \( F = \frac{5}{9}(F-32) \).

Step 2: Solve the Equation for F

Multiply both sides of the equation by 9 to eliminate the fraction: \( 9F = 5(F-32) \). Then, expand and simplify: \( 9F = 5F - 160 \). Subtract 5F from both sides: \( 4F = -160 \). Finally, divide by 4: \( F = -40 \).

Step 3: Check the Solution

Verify that \( F = -40 \) satisfies the original condition. Substitute \( F = -40 \) into the equation \( C = \frac{5}{9}(F-32) \): \( C = \frac{5}{9}(-40-32) \). Simplify: \( C = \frac{5}{9}(-72) = -40 \). The reading is indeed the same.

Step 4: Set Fahrenheit Twice the Celsius

For the second part, set the equation \( F = 2C \). Substitute \( F = 2C \) into \( C = \frac{5}{9}(F-32) \). This gives \( C = \frac{5}{9}(2C-32) \).

Step 5: Solve for C

Clear the fraction by multiplying both sides by 9: \( 9C = 5(2C - 32) \). Simplify: \( 9C = 10C - 160 \). Rearrange to get \( 160 = C \). Substitute back into \( F = 2C \): \( F = 320 \).

Step 6: Double Check the Conditions

Verify the calculations by plugging \( C = 160 \) back into the original equation: \( C = \frac{5}{9}(320-32) \). Simplifying gives \( C = 160 \), confirming the condition \( F = 2C \) holds true.

Key Concepts

Fahrenheit scale

The Fahrenheit scale is a temperature measurement system used predominantly in the United States. It was created by Daniel Gabriel Fahrenheit in the early 18th century. Unlike the Celsius scale, which is used globally, the Fahrenheit scale sets the freezing point of water at 32 degrees and the boiling point at 212 degrees. This creates a temperature range of 180 degrees between the two.

Fahrenheit scales are often used in meteorology to describe weather temperatures. This can sometimes be confusing for those accustomed to the Celsius scale, as the two systems do not align linearly. Remember when converting temperatures between scales, an accurate conversion is key to ensuring the correct value is obtained.

Understanding how the Fahrenheit scale operates can help in interpreting weather reports or cooking instructions that use this system.

Celsius scale

The Celsius scale, also known as the centigrade scale, is a temperature measurement system widely used around the world and in most scientific contexts. Developed by Anders Celsius in 1742, it is based on the freezing and boiling points of water.

• Water freezes at 0 degrees Celsius.
• Water boils at 100 degrees Celsius.

The scale divides the temperature difference between these points into 100 equal parts, which makes it easier to understand and apply.

One reason the Celsius scale is so practical is its direct alignment with the metric system, which is used internationally. In scientific research and everyday life outside the U.S., the Celsius scale is the standard for expressing temperature.

Knowing how to convert Celsius to Fahrenheit can be especially useful when engaging in international travel or scientific endeavors.

Algebraic Equations

Algebraic equations play a critical role in converting temperatures between Fahrenheit and Celsius scales. Algebra itself provides a powerful language and toolkit for solving problems that involve finding unknown values.

To convert temperatures from Fahrenheit to Celsius or vice versa, you typically set up an equation based on a known conversion formula. For instance, the formula used here is:

\[ C = \frac{5}{9}(F - 32) \]

By solving this equation, you can determine the temperature equivalent on either scale. This illustrates how equations can help find precise solutions to real-world problems, like determining the temperature where both scales match or when one scale is double the other.

Mastering algebraic techniques, like substitution and solving for variables, enables learners to navigate a variety of mathematical challenges in both academics and everyday scenarios.

Problem Solving

Problem solving involves a series of logical steps aimed at reaching a solution. In mathematics, and specifically in the exercise provided, problem solving requires understanding, setting up equations, and methodically working through them.

When addressing a temperature conversion problem:

• Identify what is known and what needs to be solved.
• Use a formula to set up the problem.
• Work through the algebraic steps to simplify the equation.
• Check the answer to ensure it satisfies the initial conditions.

In the exercise example, you must not only solve for when scale readings are equal but also when one is twice the other. This type of exercise is excellent for honing logical reasoning and mathematical skills.

Thus, mastering problem solving ensures that you can apply mathematical concepts effectively both in the classroom and in practical, real-world situations.