Question

At what temperature is the temperature in degrees Fahrenheit equal to twice the temperature in degrees Celsius?

Short answer

The temperature at which the temperature in degrees Fahrenheit is equal to twice the temperature in degrees Celsius is \(320^\circ F\).

Step-by-step solution

Write the given condition as an equation

We know that the temperature in Fahrenheit is equal to twice the temperature in Celsius. Therefore, we can write the equation: \(F = 2C\)

Substitute the conversion formula

Now, we can substitute the conversion formula for Fahrenheit into the above equation: \(\dfrac{9}{5}C + 32 = 2C\)

Solve for the Celsius temperature

To solve for C, we need to isolate it by following these steps: 1. Subtract \(\dfrac{9}{5}C\) from both sides to get all the C terms on one side: \(\begin{aligned}\dfrac{9}{5}C - \dfrac{9}{5}C +32 &= 2C - \dfrac{9}{5}C\\ 32 &= \dfrac{1}{5}C \end{aligned}\) 2. Multiply both sides by 5 to get rid of the fraction: \(\begin{aligned}5 \times 32 &= 5 \times \dfrac{1}{5}C\\ 160 &= C \end{aligned}\) So, the required temperature in degrees Celsius is \(C = 160\)

Calculate the temperature in Fahrenheit

Now, we can find the temperature in Fahrenheit using our original equation: \(F = 2C\) Substitute the value of C we found: \(F = 2 \times 160\) Calculate the value of F: \(F = 320\) Therefore, the temperature at which the temperature in degrees Fahrenheit is equal to twice the temperature in degrees Celsius is \(320^\circ F\).

Key concepts

Fahrenheit and Celsius relationship

Understanding the relationship between Fahrenheit and Celsius is key to solving temperature conversion problems. Both are units for measuring temperature, but they use different scales. In the Celsius scale, water freezes at 0 degrees and boils at 100 degrees. The Fahrenheit scale, on the other hand, has water freezing at 32 degrees and boiling at 212 degrees.
This difference in scales can cause confusion, but it's helpful to know that the two scales have a direct mathematical relationship. Essentially, as one increases or decreases, so does the other — yet not by the same numerical value. This is because the scales are offset and also have different "size" degrees.
When you're asked to find a particular relationship between the temperatures in both scales, such as when a temperature in Fahrenheit is twice that in Celsius, it helps to first understand how far apart the scales are initially, and how they progress from there.

temperature equations

Temperature equations are mathematical tools used to represent the relationship between temperatures in different scales. The exercise provided an equation that signifies a specific condition: the Fahrenheit temperature being twice the Celsius temperature. This equation, expressed as \(F = 2C\), embodies a scenario where the Fahrenheit reading is literally double the Celsius reading.
Equations like these help bridge different perspectives and calculations about temperature. By setting the two temperatures equal through mathematical operations, much like setting equal two sides of a balance, you can find a definitive point where a given condition holds true. This often requires manipulating the given equations to isolate a particular variable, as was done in the step-by-step approach of solving for Celsius first.

conversion formulas

Conversion formulas play a crucial role when converting temperatures from one scale to another, such as from Celsius to Fahrenheit. The formula \(F = \frac{9}{5}C + 32\) is used to convert a given temperature in Celsius to its equivalent in Fahrenheit.
Similarly, for converting from Fahrenheit to Celsius, the formula \(C = \frac{5}{9}(F - 32)\) is used. These formulas allow precise and consistent conversion between the two scales by accounting for both the scale differences and the shift in zero points between them.
Understanding these formulas can be helpful in solving complex problems that require manipulation or specific conditions, such as the one in the original exercise where substitution was used to introduce and solve the conversion relationship.