Question

At what temperature is the Fahrenheit scale reading equal to (a) twice that of the Celsius and \((b)\) half that of the Celsius?

Short answer

The Fahrenheit scale reading is twice the Celsius reading at 160°C. There is no positive Celsius temperature where the Fahrenheit scale reading is half the Celsius reading.

Step-by-step solution

Find Equation For (a)

First, let's find the temperature at which the Fahrenheit scale reading equals twice the Celsius reading. Based on the given, if the Fahrenheit reading is twice the Celsius reading, var t stands for the required temperature in Celsius. So write down the relationship \( F = 2C \). Set equal to the Fahrenheit to Celsius conversion formula: \( 2C = 9/5C + 32 \).

Solve Equation For (a)

Solving the equation \( 2C = 9/5C + 32 \) for C. To do this, simplify by multiplying all terms by 5 to get rid of the fraction: \( 10C = 9C + 32 \times 5 \). Then isolate the C term by subtracting 9C from both sides: \( C = 160 \). So, at 160°C, the Fahrenheit reading is twice the Celsius reading.

Find Equation For (b)

Next, let's find the temperature at which the Fahrenheit scale reading equals half the Celsius reading. Using the same approach as (a), the relationship can be written as \( F = C/2 \). Also equal to the Fahrenheit to Celsius conversion: \( C/2 = 9/5C + 32 \).

Solve Equation For (b)

Solve the equation \( C/2 = 9/5C + 32 \). Again, multiply all terms by 5 to simplify: \( 5/2C = 9C + 32 \times 5 \). Subtracting 9C from both sides to isolate the C term results in a negative Celsius temperature. Thus, there are no positive Celsius temperature values for which the Fahrenheit reading equals half the Celsius reading. So, there is no solution for this part.

Key concepts

Fahrenheit to Celsius

Converting Fahrenheit to Celsius is a common task in science and everyday life. This conversion is essential when dealing with temperature readings, as different regions and scientific studies use different temperature scales.The formula to convert a temperature from Celsius to Fahrenheit is:\[F = \frac{9}{5}C + 32\]Where:

• \(F\) represents the Fahrenheit temperature.
• \(C\) represents the Celsius temperature.

Understanding how this formula works allows us to switch between these two scales easily. It is also the basis for solving many temperature-related mathematical problems, such as comparing readings on the two scales.When you know the Celsius temperature and need to find the equivalent in Fahrenheit, just plug the Celsius value into the formula. For example, if you have 25°C, substitute 25 for \(C\) in the formula to find the Fahrenheit temperature.

Celsius temperature

The Celsius scale is a widely used temperature scale in which water freezes at 0 degrees and boils at 100 degrees under standard atmospheric conditions. Understanding Celsius is important, especially in scientific contexts. In the exercise provided, part of the problem involves finding specific Celsius values where the Fahrenheit reading is a multiple of the Celsius reading. This means manipulating the general conversion formula to express a relationship in terms of Celsius: This provides an opportunity to see Celsius not just as a standalone measurement but as part of a temperature relationship. The importance of Celsius temperatures goes beyond simple readings. It involves understanding how temperatures relate to each other and how they can be used in equations to solve for unknown variables.

Solving equations

Solving equations is a fundamental skill in mathematics, crucial for finding unknown values that satisfy a given condition. The exercise you're working on highlights how understanding and applying equations can help you find specific temperature values.When solving an equation like:\[2C = \frac{9}{5}C + 32\]The goal is to isolate the variable, in this case, \(C\), to find its specific value. The process often involves:

• Eliminating fractions by multiplying through by a common denominator.
• Rearranging terms to gather like terms on one side.
• Simplifying the equation step-by-step until the variable is isolated.

In this case, solving results in finding a specific Celsius value that meets the criteria of the problem. Recognizing that there might not always be viable solutions, such as negative values when only positive temperatures make sense, is also crucial in problem-solving.