Question

A column of liquid is found to expand linearly on heating. Assume the column rises 5.25 $\mathrm{cm}$ for a ${10.0}^{\circ }\mathrm{F}$ rise in temperature. If the initial temperature of the liquid is ${98.6}^{\circ }\mathrm{F}$ , what will the final temperature be in ${}^{\circ }\mathrm{C}$ if the liquid has expanded by 18.5 $\mathrm{cm}$ ?

Short answer

The final temperature of the liquid in Celsius is $38.7994°C$.

Step-by-step solution

Calculate the rise in temperature in Fahrenheit

First, let's find out the rise in temperature in Fahrenheit when the liquid expands by 18.5 cm. We know that the column rises 5.25 cm for a 10.0°F rise in temperature. Let the rise in temperature in Fahrenheit be x °F, then we can set up a proportion: $\frac{5.25cm}{10.0°F}=\frac{18.5cm}{x°F}$ Now, we will solve for x.

Solve the proportion

To solve the proportion, we can cross-multiply and solve for x: $5.25°F\ast x=10.0°F\ast 18.5cm$ Divide both sides of the equation by 5.25 °F to isolate x: $x=\frac{10.0°F\ast 18.5cm}{5.25°F}$ Now, calculate the rise in temperature (x): $x=35.2381°F$ Now we know that the rise in temperature is 35.2381°F.

Convert the initial temperature to Celsius

The initial temperature in Fahrenheit is given as $98.6°F$. To convert it to Celsius, we will use the temperature conversion formula: $°C=(°F-32)\ast \frac{5}{9}$ Plug the initial temperature (98.6°F) into the formula: Initial Temperature in Celsius = $(98.6°F-32)\ast \frac{5}{9}$ Calculate the initial temperature in Celsius: Initial Temperature in Celsius = $37°C$

Calculate the final temperature in Celsius

Now we have the initial temperature in Celsius and the rise in temperature in Fahrenheit. First, we need to convert the rise in temperature in Fahrenheit to Celsius using the same temperature conversion formula: Rise in temperature in Celsius = $(35.2381°F-32)\ast \frac{5}{9}$ Calculate the rise in temperature in Celsius: Rise in Temperature in Celsius = $1.7994°C$ Now add the initial temperature and the rise in temperature to get the final temperature: Final Temperature in Celsius = $37°C+1.7994°C$ Calculate the final temperature: Final Temperature in Celsius = $38.7994°C$ So, the final temperature of the liquid in Celsius is $38.7994°C$.

Key concepts

Temperature Conversion

Temperature conversion is an essential concept in understanding the relationship between different temperature scales. This process is vital in scientific computations and everyday applications. Commonly used temperature scales include Fahrenheit, Celsius, and Kelvin, with conversions often necessary between them.

To convert a temperature from Fahrenheit to Celsius, we use the formula:

• Calculate the difference from the baseline (32° F for freezing point of water).
• Multiply the resulting value by $\frac{5}{9}$ to convert the unit scale.

For instance, converting from 98.6° F, a commonly known average human body temperature in Fahrenheit, to Celsius involves subtracting 32 and then multiplying the result by $\frac{5}{9}$. The conversion provides a Celsius equivalent, essential for scientific calculations where Celsius is often a preferred unit.

Understanding this basic formula allows one to skip confusion between temperature readings in various geographical locations or scientific setups, where either Celsius or Fahrenheit might be used by default.

Linear Expansion

Linear expansion refers to the increase in length or volume of a material when it is subjected to a change in temperature. This principle is widely observed in liquids, solids, and even gases when considering volume expansion.

In the context of this exercise, the linear expansion of the liquid column is linearly dependent on the amount of temperature increase. This observation is critical in practical applications like construction and engineering, where temperature variations can significantly affect material dimensions.

Mechanically, this is described by the formula:

• $\text{Change in Length}=\text{initial length}\times \text{coefficient of linear expansion}\times \text{change in temperature}$

This fundamental formula is simplified in the exercise as the exact coefficient isn't required; rather, the exercise uses known expansion data to develop a simple proportion to calculate the change in temperature required for a given expansion in length.

Proportion in Mathematics

Proportions in mathematics help to solve problems where quantities change in relation to each other. This principle applies when calculating the amount of temperature change leading to a certain degree of expansion.

In this exercise, proportions are used to relate the change in the length of a liquid column to the increase in temperature. The problem states that a 5.25 cm rise in the column corresponds to a 10°F increase in temperature. When solving:

• The ratio $\frac{5.25\text{ cm}}{10.0\text{ °F}}$ is used to establish a proportional relationship with the expanded column length of 18.5 cm.
• Cross multiply to find the unknown rise in temperature in °F.
  

This approach provides a straightforward mathematical model to handle scenarios where direct measurement, using coefficients or equations, is not feasible or necessary, making proportionate reasoning a powerful tool in problem-solving.

Fahrenheit to Celsius Conversion

Converting temperature changes between Fahrenheit and Celsius accurately is crucial, especially in scientific and engineering contexts where precision matters. The need to convert rise in temperature from Fahrenheit to Celsius is addressed using the standard conversion formula.

• First, calculate the change in Fahrenheit, which in this example was found to be 35.2381°F.
• Then apply the conversion: $\text{Change in °C}=(\text{Change in °F}-32)\times \frac{5}{9}$.

This is slightly more complex than a straightforward absolute temperature conversion because it handles differences, but the principle is the same except we don't subtract 32 for the difference itself.

This ensures that calculations maintain accuracy across differing temperature scales and supports computations necessary for experimental and real-life applications.