Problem 53 - Elementary Algebra

Question

Set up an algebraic inequality and then solve it. A computer is set to shut down if the temperature exceeds \(40^{\circ} \mathrm{C}\). Give an equivalent statement using degrees Fahrenheit. (Hint: \(C=59(F-32) .)\)

Step-by-Step Solution

Answer

The computer shuts down if the temperature exceeds 104°F.

Step 1: Identify the Inequality Condition

The problem requires us to find at what Fahrenheit temperature a computer shuts down if it exceeds 40°C. This can be translated to the inequality condition: \( C > 40 \).

Step 2: Use the Conversion Formula

We are given the conversion formula \( C = \frac{5}{9} (F - 32) \). We will use this formula to express the Celsius temperature limit in terms of Fahrenheit.

Step 3: Substitute the Celsius Limit

Substitute \( C = 40 \) into the conversion formula: \[ 40 = \frac{5}{9} (F - 32) \].

Step 4: Solve for F

To solve for \( F \), multiply both sides by 9 to eliminate the fraction:\[ 360 = 5(F - 32) \]. Then, divide both sides by 5:\[ 72 = F - 32 \]. Finally, add 32 to both sides:\[ F = 104 \].

Step 5: Establish the Equivalent Inequality

Thus, an equivalent inequality in Fahrenheit to the condition \( C > 40 \) is \( F > 104 \).

Key Concepts

Temperature Conversion

Temperature conversion is essential for understanding how to switch from one temperature scale to another. This concept is vital in many scientific and industrial applications where temperature is a key parameter.
There are various temperature scales, but the most common ones are Celsius and Fahrenheit. In temperature conversion, we use formulas to shift values from one scale to another.

• Celsius (\(^{ m{o}} C \)) is commonly used around the world and in scientific communities.
• Fahrenheit (\(^{ m{o}} F \)) is mainly used in the United States for everyday temperature readings.

Understanding temperature conversion enables us to interpret weather reports, scientific data, and operate temperature-sensitive equipment correctly. Let's delve deeper into converting between Fahrenheit and Celsius, a frequent task in such applications.

Fahrenheit to Celsius Conversion

The conversion between Fahrenheit and Celsius involves a specific formula. This formula allows us to translate temperature from one scale to the other. The conversion formula is \( C = \frac{5}{9} (F - 32) \), where:

• \( C \) is the temperature in Celsius.
• \( F \) is the temperature in Fahrenheit.

The formula translates a Fahrenheit temperature into Celsius by accounting for the fact that the two scales have different zero points and intervals.
The steps are as follows:

• Subtract 32 from the Fahrenheit temperature,
• Multiply the result by \(\frac{5}{9}\).

This is handy in contexts where a given temperature must be precisely converted, such as ensuring safe operating conditions for electronic equipment.

Algebraic Equations

Algebraic equations are fundamental in defining relationships between different quantities. They consist of variables and constants and are connected by an equal sign.
These equations can represent real-world scenarios, such as computing a temperature limit that dictates machine operations.
In the context of temperature, we encounter algebraic equations when converting temperatures or expressing conditions in different units. The conversion of temperature from Fahrenheit to Celsius in the original example is expressed as:
\[ C = \frac{5}{9} (F - 32) \]
This equation helps us understand the relationship between Celsius and Fahrenheit. It also showcases how a simple algebraic approach can interpret everyday conditions, reinforcing the importance of algebra in practical situations.

Solving Inequalities

Solving inequalities involves finding the range of values that satisfy a given inequality condition. Unlike algebraic equations, inequalities use symbols such as \(<, >, \leq,\) and \(\geq\).
In real-world applications, like the exercise above, inequalities can set conditions for safe operations, like ensuring a computer does not overheat. In our example, we determined when the computer should shut down in Fahrenheit if it exceeds \(40^{\circ} C\).
To proceed, the inequality \(C > 40\) was converted into Fahrenheit: \[ F > 104 \]
This inequality tells us that the Fahrenheit temperature must remain under \(104^{\circ} F\) to prevent a shutdown. Solving inequalities involves steps similar to solving equations but with attention to maintaining the inequality direction. This method is essential for setting limits and thresholds in various technological and scientific fields.