Problem 1 - Discrete Mathematics with Applications

Question

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). Express \(-40^{\circ} \mathrm{C}\) on the Fahrenheit scale.

Step-by-Step Solution

Answer

The equivalent temperature of \(-40^{\circ} \mathrm{C}\) on the Fahrenheit scale is \(-40^{\circ} \mathrm{F}\).

Step 1: Identify the given equation and values

We are given the formula to convert Celsius to Fahrenheit: \[F = \frac{9}{5} C + 32\] We are also given the Celsius temperature: \[C = -40^{\circ}\]

Step 2: Substitute the given Celsius value into the formula

Now we can substitute the given Celsius temperature into the formula: \[F = \frac{9}{5} (-40) + 32\]

Step 3: Solve for F

Next, we need to solve for the Fahrenheit temperature F: \[F = \frac{9}{5}(-40) + 33 = \frac{(-360)}{5} + 32\] Now, divide -360 by 5: \[-72 = -360/5\] Finally, add 32 to -72: \[-40 = -72 + 32\]

Step 4: Write the answer

The equivalent temperature on the Fahrenheit scale is -40°F. Therefore, -40°C is equal to -40°F.

Key Concepts

Celsius to Fahrenheit

Understanding how to convert temperatures from Celsius to Fahrenheit is a fundamental skill in science and everyday life. The formula for this conversion is given by \( F = \frac{9}{5} C + 32 \)
where \( F \) represents the temperature in degrees Fahrenheit and \( C \) denotes the temperature in degrees Celsius.

Breaking Down the Conversion

To grasp this formula better, let's dissect it:

• The ratio \( \frac{9}{5} \) is the key to converting the temperature scales, as it accounts for the difference in degree increments between the two systems.
• The addition of 32 to the result adjusts for the offset in the two scale's zero points.

When you're presented with a Celsius temperature, like \( -40^\circ C \), apply these steps to calculate the Fahrenheit equivalent. Multiply the Celsius temperature by \( \frac{9}{5} \) and then add 32. Continue with this straightforward arithmetic, and you'll have converted the temperature with ease.

Algebraic Problem Solving

Algebraic problem solving involves using variables to represent unknowns and performing operations to solve for these variables. Temperature conversion problems, like the one in our example, provide a perfect context for practicing algebraic techniques.
In the given problem, you resolved for \( F \) by following algebraic steps:

Substitution

• Replace known values into the appropriate places of the equation.

Arithmetic Operations

• Perform multiplication and addition as per order of operations.

Reaching the Solution

• Arrive at the final answer through calculation. It's crucial to keep track of positive and negative signs, especially when dealing with temperatures below zero.

By mastering these algebraic methods, you can solve a vast range of problems beyond temperature conversion, enhancing your mathematical toolkit.

Temperature Scales

Temperature scales are essential for understanding and communicating thermal conditions. The most commonly used scales are Celsius and Fahrenheit.

Celsius Scale

The Celsius scale is favored in scientific and most international contexts. It sets the freezing point of water at \(0^\circ C\) and the boiling point at \(100^\circ C\), under standard atmospheric pressure.

Fahrenheit Scale

The Fahrenheit scale is commonly used in the United States. Here, water freezes at \(32^\circ F\) and boils at \(212^\circ F\).
Understanding these scales enables us to accurately describe temperatures in different settings and convert between scales when needed. Real-world applications range from weather forecasting and culinary arts to science labs and healthcare.