Problem 82 - Algebra and Trigonometry

Question

Temperature Scales What interval on the Celsius scale corresponds to the temperature range \(50 \leq F \leq 95 ?\)

Step-by-Step Solution

Answer

The Celsius interval corresponding to 50°F to 95°F is 10°C to 35°C.

Step 1: Understand the Relationship

The relationship between the Fahrenheit and Celsius scales is given by the formula: \[ C = \frac{5}{9}(F - 32) \] where \( C \) is the temperature in Celsius and \( F \) is the temperature in Fahrenheit. We will use this formula to convert the given Fahrenheit interval to Celsius.

Step 2: Convert the Lower Bound

We first convert 50°F to Celsius:\[ C_1 = \frac{5}{9}(50 - 32) \]\[ C_1 = \frac{5}{9} \times 18 \]\[ C_1 = 10 \]Thus, 50°F corresponds to 10°C.

Step 3: Convert the Upper Bound

Next, convert 95°F to Celsius:\[ C_2 = \frac{5}{9}(95 - 32) \]\[ C_2 = \frac{5}{9} \times 63 \]\[ C_2 = 35 \]Thus, 95°F corresponds to 35°C.

Step 4: Determine the Celsius Interval

The interval on the Celsius scale is from the lower bound of 10°C to the upper bound of 35°C. Therefore, the temperature range 50°F to 95°F corresponds to the Celsius interval of \(10 \leq C \leq 35\).

Key Concepts

Temperature Conversion

Temperature conversion is the process of changing the value of temperature from one scale to another. This fundamental concept in science and everyday life helps us adapt to different measurement systems that are used globally. The most commonly used temperature scales are Celsius (°C) and Fahrenheit (°F).

Understanding the conversion formulas is critical in order to precisely interpret temperatures across these scales. The formula for converting Fahrenheit to Celsius is:

• \( C = \frac{5}{9}(F - 32) \)

This formula tells us how to convert any temperature value in Fahrenheit into its equivalent in the Celsius scale. When converting, subtract 32 from the Fahrenheit temperature, multiply the result by 5, and then divide by 9. This formula is based on the difference in the freezing and boiling points of water on the two scales, and it's essential for accurate temperature conversion.

Fahrenheit to Celsius

Converting temperatures from Fahrenheit to Celsius involves using a specific formula that accounts for the different starting points and increments of these scales. The Fahrenheit scale sets the freezing point of water at 32°F and boiling point at 212°F.

In contrast, the Celsius scale places these at 0°C and 100°C, respectively. This difference affects how we convert between these two temperature measurements.

For example, to convert a temperature range like 50°F to 95°F into Celsius, we apply the conversion formula for both ends of the range:

• For 50°F: \( C_1 = \frac{5}{9}(50 - 32) = 10°C \)
• For 95°F: \( C_2 = \frac{5}{9}(95 - 32) = 35°C \)

The conversion results in a temperature range of 10°C to 35°C, which corresponds to the initial Fahrenheit range. Understanding these conversion methods and dry numerical calculations can be important for scientific experiments, cooking, and travel.

Temperature Scales

Temperature scales play an important role in measuring how hot or cold an environment or object is. The two most prevalent temperature scales are Celsius, commonly used around the world, and Fahrenheit, primarily used in the United States.

Each scale has unique characteristics. The Celsius scale is based on the physical properties of water, with 0°C as the freezing point and 100°C as the boiling point. The Fahrenheit scale, designed by Daniel Gabriel Fahrenheit, is more tailored to human experience with 32°F as the freezing point and 212°F as the boiling point for water.

Aside from these commonly used scales, there are others like Kelvin and Rankine, primarily used in scientific applications. However, understanding Celsius and Fahrenheit is vital as they are the most frequent in day-to-day temperature readings, climatic conditions, and weather forecasts. These temperature scales allow for clear communication and understanding in both scientific and daily life scenarios.