Question

At what temperature the centigrade (celsius) and Fahrenheit readings at the same. \((\mathrm{A})-40^{\circ}\) (B) \(+40^{\circ} \mathrm{C}\) (C) \(36.6^{\circ}\) (D) \(-37^{\circ} \mathrm{C}\)

Short answer

The temperature at which the Celsius and Fahrenheit readings are the same is \(-40^{\circ}\).

Step-by-step solution

Conversion formula

The formula to convert a Celsius temperature to Fahrenheit is \(F = \frac{9}{5}C + 32\) where F is the temperature in Fahrenheit, and C is the temperature in Celsius. #Step 2: Set the Celsius and Fahrenheit temperatures equal to each other#

Setting the temperatures equal

We are asked to find the temperature at which the Celsius and Fahrenheit readings are the same. So, we need to find the value of \(C\), the Celsius temperature, so that \(C=F\) Therefore, we will substitute the conversion formula into the equation: \(C = \frac{9}{5}C + 32\) #Step 3: Solve the equation for the Celsius temperature#

Solving the equation

To solve the equation, we first need to get the same denominator for both terms: \(5C = 9C + 160\) Then we will subtract \(9C\) from both sides of the equation: \(-4C = 160\) Finally, we will divide both sides by \(-4\): \(C = -40\) #Step 4: Check the solution in the options given#

Checking the solution

We found that the Celsius temperature at which the Celsius and Fahrenheit readings are the same is: \(C = -40^{\circ}\) Now, we need to check the options given: (A) \(-40^{\circ}\) (Correct) (B) \(+40^{\circ} \mathrm{C}\) (Not correct) (C) \(36.6^{\circ}\) (Not correct) (D) \(-37^{\circ} \mathrm{C}\) (Not correct) The answer is (A), \(-40^{\circ}\).

Key concepts

Celsius and Fahrenheit equivalence

Celsius and Fahrenheit are two temperature scales that find use in different regions and contexts. The Celsius scale, often used in scientific settings and most countries worldwide, sets the freezing point of water at 0°C and the boiling point at 100°C. Meanwhile, the Fahrenheit scale, primarily used in the United States, sets these points at 32°F and 212°F, respectively.

At the core of this exercise is the question of when the temperature readings in Celsius and Fahrenheit are the same. This equivalence happens at a single point: \[-40\]. Therefore, at \(-40^\circ\), both Celsius and Fahrenheit scales register the same value, a rare coincidence in the mathematical relationship between the two scales. This occurs due to the unique linear equation governing the conversion between the two scales.

Linear equations in physics

Linear equations are vital tools in physics for describing linear relationships between quantities. In the context of temperature conversion, the relationship between Celsius and Fahrenheit is modeled by a linear equation. The equation used to convert Celsius to Fahrenheit is \(F = \frac{9}{5}C + 32\). This equation shows how temperatures in Celsius convert directly into Fahrenheit using a straightforward linear relationship.

In this equation, \(\frac{9}{5}\) is the slope, indicating how many degrees Fahrenheit change for each degree Celsius, while 32 is the intercept, signifying the base temperature in Fahrenheit when Celsius is zero.

Setting the Celsius and Fahrenheit values equal \(C=F\), introduces an opportunity to solve the equation to find where both scales equal the same number. This unique exercise highlights the usefulness of linear equations in solving real-world problems by simply rearranging and solving the equation.

Temperature scales

Temperature scales are systems that measure temperature, and they are based on chosen reference points. The Celsius scale, a metric measurement system, uses the freezing and boiling points of water to establish 0°C and 100°C as measurement points. Its counterpart, the Fahrenheit scale, uses 32°F for water's freezing point and 212°F for boiling. These differences in reference points cause readings to vary significantly between the scales at any given temperature.

Knowing how to convert between temperature scales is essential, as it helps in understanding weather conditions, scientific data, and everyday temperatures across different regions. This exercise demonstrates the practical application of temperature scales and their conversion, underscoring the importance of having a standard method to translate these references accurately.

By understanding temperature scales, we gain better insight into how we experience temperature in various aspects of life and the relationships between different units of temperature measurement, such as Kelvin, Rankine, and others, though the primary focus here is on Celsius and Fahrenheit.