Problem 74 - 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 interval is \(10 \leq C \leq 35\).

Step 1: Understand the Problem

We need to convert the temperature range from Fahrenheit (F) to Celsius (C) and determine the interval in Celsius.

Step 2: Use the Conversion Formula

The formula to convert Fahrenheit to Celsius is \( C = \frac{5}{9}(F - 32) \), where \( C \) is the temperature in Celsius and \( F \) is the temperature in Fahrenheit.

Step 3: Convert Lower Bound

Convert the lower bound of the interval by substituting \( F = 50 \) into the formula:\[ C_1 = \frac{5}{9}(50 - 32) = \frac{5}{9} \times 18 = 10 \]So, the lower bound on the Celsius scale is 10°C.

Step 4: Convert Upper Bound

Convert the upper bound of the interval by substituting \( F = 95 \) into the formula:\[ C_2 = \frac{5}{9}(95 - 32) = \frac{5}{9} \times 63 = 35 \]So, the upper bound on the Celsius scale is 35°C.

Step 5: Write the Interval

With the calculations complete, the interval on the Celsius scale that corresponds to the Fahrenheit interval \(50 \leq F \leq 95\) is \(10 \leq C \leq 35\).

Key Concepts

Temperature Scales

Temperature scales are essential in everyday life and scientific work. They allow us to measure and compare thermal energy.

• The most commonly used temperature scales are Celsius, Fahrenheit, and Kelvin.
• The Celsius scale is based on the freezing and boiling points of water, set at 0°C and 100°C, respectively.
• The Fahrenheit scale sets the freezing point of water at 32°F and the boiling point at 212°F.

Different temperature scales can lead to confusion, so it's crucial to understand how to convert between them. Understanding these scales allows us to interpret and compare temperatures in different contexts, whether we’re checking the weather or performing a scientific experiment.

Algebraic Conversion

Algebraic conversion is the process of using a formula to transform data from one system to another. In the case of temperature conversions, we use algebraic equations to switch between Fahrenheit and Celsius scales.
The conversion formula between these two scales is:

• For Fahrenheit to Celsius: \( C = \frac{5}{9} (F - 32) \)
• For Celsius to Fahrenheit: \( F = \frac{9}{5} C + 32 \)

These formulas involve simple arithmetic. Subtraction, multiplication, and division help in accurately converting temperatures. It's important to apply these calculations correctly to ensure that you change the units without altering the temperature's meaning. With practice, these conversions become quick and intuitive, making it easier to navigate between different temperature scales.

Celsius Scale

The Celsius scale, also known as the centigrade scale, is widely used around the world for scientific temperature measurement. Named after the Swedish astronomer Anders Celsius, it is based on water's properties.

• The freezing point of water is defined as 0°C.
• The boiling point of water is defined as 100°C at standard atmospheric pressure.

This range is divided into 100 equal parts, called degrees Celsius. This straightforward division makes the Celsius scale simple and intuitive. It aligns closely with physical properties and everyday experiences, making it a practical choice for education and daily use. In scientific contexts, the Celsius scale is often used alongside the Kelvin scale, which begins at absolute zero (0 K = -273.15°C), to measure temperature shifts and processes.