Question

A Fahrenheit and a Celsius thermometer are immersed in the same medium. At what Celsius temperature will the numerical reading on the Fahrenheit thermometer be (a) \(49^{\circ}\) less than that on the Celsius thermometer; (b) twice that on the Celsius thermometer; (c) one-eighth that on the Celsius thermometer; (d) \(300^{\circ}\) more than that on the Celsius thermometer?

Short answer

(a) The Celsius temperature at which it will be 49 degrees less on the Fahrenheit thermometer is -59 degrees. (b) The Celsius temperature at which it will be twice on the Fahrenheit thermometer is 160 degrees. (c) The Celsius temperature at which it will be one-eighth on the Fahrenheit thermometer is -176.8 degrees. (d) The Celsius temperature at which it will be 300 degrees more on the Fahrenheit thermometer is 135 degrees.

Step-by-step solution

Understand the Conversion Formula

The first thing to remember is that the conversion from a temperature from Celsius to Fahrenheit is given by the formula: \(F = \frac{9}{5}C + 32\). In this formula, \(F\) represents the temperature in Fahrenheit and \(C\) represents the temperature in Celsius.

Formulate and Solve for (a)

(a) From the given condition, we know that the Celsius temperature is given as \(F + 49\). We substitute this into the conversion formula:\[\frac{9}{5}(F + 49) + 32 = F\]Solving this equation, we find \(C = -59\degree\)

Formulate and Solve for (b)

(b) For this part, we need the reading on the Fahrenheit scale to be twice that on the Celsius scale. This gives us the equation:\[\frac{9}{5}C + 32 = 2C\]Solving this equation, we find \(C = 160\degree\)

Formulate and Solve for (c)

(c) The condition here is that the Celsius temperature is eight times the Fahrenheit temperature. Hence, the equation is:\[\frac{9}{5}C + 32 = \frac{C}{8}\]Solving this equation, we find \(C = -176.8\degree\)

Formulate and Solve for (d)

(d) The final condition is that the Fahrenheit temperature is \(300\degree\) more than the Celsius temperature. The equation is:\[\frac{9}{5}C + 32 = C + 300\]Solving this equation, we find \(C = 135\degree\)

Key concepts

Fahrenheit to Celsius

When working with temperature conversions, one common task is to convert a temperature reading from Fahrenheit to Celsius. This is particularly useful when traveling or working with scientific data that often uses Celsius. The formula used for this conversion is:- \[ C = \frac{5}{9}(F - 32) \]- In this equation, \(C\) stands for the temperature in Celsius, and \(F\) is the temperature in Fahrenheit. The subtraction of 32 adjusts for the difference in the starting points of the two temperature scales. Then, the multiplication by \(\frac{5}{9}\) converts the Fahrenheit unit to a Celsius unit, which is smaller.
Understanding this formula is crucial because it allows us to easily switch between two widely used temperature scales. This skill is particularly important when solving problems like those seen in the exercise, where relationships between Celsius and Fahrenheit readings are explored. Always start by writing the formula correctly and substituting the given Fahrenheit value to find the Celsius temperature. This method provides clarity and precision in temperature conversions.

Temperature Equations

Temperature conversion often involves setting up equations, especially when conditions compare the Fahrenheit reading with the Celsius reading. For example, suppose a problem states that the Fahrenheit temperature is 49 degrees less than the Celsius temperature. In that case, we express this relationship with:- \[ C = F + 49 \]- Using the conversion formula \(F = \frac{9}{5}C + 32\), substitute \(F\) in terms of \(C\) into the relationship to form an equation.
These equations may also involve inequalities or ratios. For instance, if a Fahrenheit temperature is twice the Celsius, the equation would be:- \[ F = 2C \]- Substitute to convert this into a solvable form. This approach simplifies the process of finding Celsius when given conditions change the way temperatures relate to one another.
Equations like these are not only mathematical exercises but practical skills that can assist in fields such as meteorology, climate science, and even everyday cooking where setting the right temperature is key.

Celsius to Fahrenheit

Converting a Celsius temperature to Fahrenheit involves using the conversion formula in reverse, but this time aiming for a Fahrenheit result. The formula is:- \[ F = \frac{9}{5}C + 32 \]- This expression shows that each degree Celsius translates into a larger degree Fahrenheit unit after multiplying by \(\frac{9}{5}\), and then adding 32 compensates for the starting point difference between the two scales.
Utilizing this formula becomes particularly helpful when you need to understand temperatures in the context of weather forecasts given in Celsius while being in a place where Fahrenheit is used. Similarly, in scientific experiments under laboratory conditions, converting to Fahrenheit helps in understanding real-world applications requiring exact temperature controls.
An understanding of such conversions is crucial, especially because the conversion formulas remain constant regardless of external conditions. Practice using the formula by plugging in Celsius values to develop a stronger intuitive grasp of how Celsius temperatures manifest in the Fahrenheit scale.