Problem 120 - Algebra and Trigonometry

Question

The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F,\) is $$ F=\frac{9}{5} C+32 $$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Step-by-Step Solution

Answer

The equivalent Fahrenheit temperature range for the given Celsius range is [59, 95]

Step 1: Identify the provided range in Celsius

The range provided is from 15°C to 35°C inclusive.

Step 2: Apply the conversion formula for the lower limit

Apply the conversion formula \(F = \frac{9}{5}C + 32\) for the lower limit of the Celsius range, which is 15. So, \(F_1 = \frac{9}{5}*15 + 32 = 59\)

Step 3: Apply the conversion formula for the upper limit

Next, apply the conversion formula \(F = \frac{9}{5}C + 32\) for the upper limit of the Celsius range, which is 35. So, \(F_2 = \frac{9}{5}*35 + 32 = 95\).

Step 4: Express the range in interval notation

Finally, express the Fahrenheit range in interval notation. The interval is \([F_1, F_2]\) or \([59, 95]\) .

Key Concepts

Temperature Conversion

Temperature conversion is a process used to change the value of a temperature from one unit to another. The most common temperature scales are Celsius (°C) and Fahrenheit (°F). To convert temperatures from Celsius to Fahrenheit, a specific formula is used: \[\begin{equation} F = \frac{9}{5} C + 32 \end{equation}\]
This formula reflects that Fahrenheit and Celsius scales have different zero points and scale increments. The freezing point of water is 0 degrees Celsius or 32 degrees Fahrenheit, while the boiling point is 100 degrees Celsius or 212 degrees Fahrenheit. Each Celsius degree is equivalent to 1.8 Fahrenheit degrees. Understanding this conversion is crucial for many scientific calculations, cooking recipes, and even weather forecasting. The formula provides a linear relationship between the two scales and allows for a seamless transition from one to another.

Interval Notation

Interval notation is a mathematical representation used to express the range of values between two endpoints. It is commonly used to denote ranges of numbers, including temperatures. In interval notation, square brackets \[\begin{equation}[a, b]\end{equation}\]
indicate that the range includes the endpoints (a and b), which is known as a closed interval. Alternatively, parentheses \[\begin{equation}(a, b)\end{equation}\]
indicate that the endpoints are not included, known as an open interval. For temperature ranges, including those converted from Celsius to Fahrenheit, interval notation provides a clear, concise way to represent the minimum and maximum values inclusively or exclusively. When working with temperature conversions, using interval notation can depict converted temperature ranges clearly, which is especially useful in scientific and engineering contexts.

Applying Formulas

Applying formulas is a systematic method of using mathematical formulations to calculate or evaluate a particular quantity. When approaching a problem involving temperature conversion, the steps involve identifying the initial data, selecting the appropriate formula, and substituting the data into the formula to find the solution. To convert a range of temperatures from Celsius to Fahrenheit, two applications of the conversion formula are required:

• One for the lower limit of the Celsius range
• One for the upper limit of the Celsius range

This ensures that the entire interval of temperatures is accurately represented in the new unit of measure. The key to correctly applying any formula lies in understanding the variables involved and performing the mathematical operations with precision. This skill not only helps with converting temperatures but is a foundational ability in all scientific and mathematical endeavors, from calculating areas and volumes to determining the speed of an object.