Question

At what temperature do the Celsius and Fahrenheit temperature scales have the same numeric value? a) -40 degrees b) 0 degrees c) 40 degrees d) 100 degrees

Short answer

a) -40 degrees b) 0 degrees c) 32 degrees d) 100 degrees Answer: a) -40 degrees

Step-by-step solution

Write down the equation for Celsius to Fahrenheit conversion.

The equation to convert Celsius to Fahrenheit is F = (9/5)C + 32.

Set Celsius value (C) equal to Fahrenheit value (F).

Since we are looking for the temperature at which both Celsius and Fahrenheit have the same numeric value, we will set the Fahrenheit equation to C. This gives us the equation C = (9/5)C + 32.

Solve the equation for C.

To solve for C, we will first subtract (9/5)C from both sides of the equation. This will simplify the equation to: C - (9/5)C = 32. Now, we will find a common denominator for the left side of the equation. Since the common denominator for 1 and 5 is 5, we multiply C by (5/5) to get: (5/5)C - (9/5)C = 32. Now, we will combine the terms on the left side of the equation: (-4/5)C = 32. Now, to solve for C, we will divide both sides of the equation by (-4/5). This gives us: C = 32 / (-4/5).

Calculate the temperature value.

To calculate the temperature value, we will simplify the right side of the equation by multiplying both sides by the reciprocal of (-4/5): C = 32 * (-5/4). This simplifies to: C = -40.

Compare the answer to the given options.

Our calculated temperature value is -40 degrees, which corresponds to option a) -40 degrees. Therefore, the temperature at which Celsius and Fahrenheit scales have the same numeric value is -40 degrees.

Key concepts

Celsius to Fahrenheit Conversion

In the realm of temperature measurements, converting from Celsius to Fahrenheit is a common task, and understanding the formula is essential. The conversion equation is given by \( F = \frac{9}{5}C + 32 \), where 'F' represents degrees Fahrenheit and 'C' represents degrees Celsius. This formula shows that for each degree Celsius, there is a corresponding temperature in degrees Fahrenheit, which is found by multiplying the Celsius temperature by 9/5 and then adding 32. For example, to convert 20 degrees Celsius to Fahrenheit, the calculation would be \( F = \frac{9}{5}\times20 + 32 = 68 \) degrees Fahrenheit.

When given an exercise to find a temperature that is the same in both Celsius and Fahrenheit, we employ the same equation, but we look for the point where \( C = F \), indicating the scales intersect.

Temperature Scales Comparison

The Celsius and Fahrenheit scales are the most commonly used temperature scales but measure temperature differently. The Celsius scale is based on the boiling and freezing points of water at sea level, set at 100 degrees and 0 degrees, respectively. In contrast, the Fahrenheit scale sets the water's freezing point at 32 degrees and the boiling point at 212 degrees. A comparison of the two scales highlights that Celsius measurements are larger in magnitude per degree change compared to Fahrenheit.

It's noteworthy to mention the exact temperature where the Celsius and Fahrenheit scales read the same value, which is -40 degrees. This point of intersection can be found using the conversion formula and algebraic methods to solve for the equivalent temperature.

Solving Algebraic Equations

Successfully solving algebraic equations is pivotal in a wide array of scientific computations, including temperature conversion. In the problem of finding where Celsius and Fahrenheit scales have the same value, we manipulate the conversion formula to solve for 'C'. Starting with the equation \( C = \frac{9}{5}C + 32 \), we look to isolate 'C' on one side of the equation. We first find a common denominator and combine like terms, resulting in \( (5/5)C - (9/5)C = 32 \). This simplifies down to \( (-4/5)C = 32 \). Finally, we multiply both sides by the reciprocal of -4/5 to solve for 'C' and find that the temperature at which the scales match is -40 degrees Celsius.

Solving these equations requires careful manipulation of fractions and variables—a fundamental skill in both mathematics and physics.

Units of Temperature

Temperature is one of the most frequently measured physical properties, and various units exist to represent it. The three primary units are Celsius (°C), Fahrenheit (°F), and Kelvin (K). Kelvin is the base unit of temperature in the International System of Units (SI) and is used extensively in the scientific community. Unlike Celsius and Fahrenheit, which can have positive or negative values, the Kelvin scale starts at absolute zero, the lowest theoretically attainable temperature.

Understanding the differences in units is critical when working with temperature data, especially in scientific experiments and when comparing climate data across different regions. Each unit has a place, whether it's the everyday reporting of weather in Fahrenheit or Celsius or the precise measurements required in scientific research using Kelvin.