Problem 63 - College Algebra with Corequisite Support

Question

In chemistry the volume for a certain gas is given by \(V=20 T\), where \(V\) is measured in cc and \(T\) is temperature in \({ }^{\circ} \mathrm{C}\). If the temperature varies between \(80^{\circ} \mathrm{C}\) and \(120^{\circ} \mathrm{C}\), find the set of volume values.

Step-by-Step Solution

Answer

The volume values range from 1600 cc to 2400 cc.

Step 1: Understanding the Relationship

We are given an equation for the volume of a gas, which is dependent on temperature: \( V = 20T \). Here, \( V \) is the volume in cubic centimeters (cc) and \( T \) is the temperature in degrees Celsius (\(^{\circ} C\)).

Step 2: Identifying the Range of Temperature

The temperature \( T \) is said to vary between \(80^{\circ} \mathrm{C}\) and \(120^{\circ} \mathrm{C}\). This means \( T \) can take any value between 80 and 120, inclusive.

Step 3: Calculating Volume at the Lower Temperature Limit

Substitute the lower limit of the temperature \( T = 80^{\circ} \mathrm{C} \) into the equation. Then, \( V = 20 \times 80 = 1600 \) cc.

Step 4: Calculating Volume at the Upper Temperature Limit

Substitute the upper limit of the temperature \( T = 120^{\circ} \mathrm{C} \) into the equation. Then, \( V = 20 \times 120 = 2400 \) cc.

Step 5: Formulating the Set of Volume Values

Since the volume \( V \) increases linearly with temperature \( T \), the values for \( V \) will vary continuously between 1600 cc and 2400 cc as \( T \) varies between 80 and 120. Hence, the set of volume values is \( \{ V \mid 1600 \leq V \leq 2400 \} \).

Key Concepts

Volume of Gas

Understanding the volume of a gas involves recognizing how it relates directly to temperature. In our equation, we have the formula \( V = 20T \), where \( V \) represents volume in cubic centimeters (cc) and \( T \) is the temperature in degrees Celsius. This equation shows a linear relationship. This means if you double the temperature, the volume also doubles. This is important because it simplifies predictions and calculations regarding the behavior of gases under different conditions.

• Volume is directly proportional to temperature.
• Every change in temperature has a magnitude effect on the volume.

Given this linear relationship, you can determine the volume at any given temperature by simple multiplication. This is fundamental in chemistry and physics where precise measurements and predictions are necessary.

Temperature Range

The temperature range is given as 80°C to 120°C. In this context, a range refers to the set of possible values that a variable can take. When using the equation \( V = 20T \), any temperature within this range can be substituted directly to find the corresponding volume. Linear equations like these make things straightforward because changes are predictable and consistent.

• Ranging from 80°C to 120°C gives a clear boundary for calculations.
• Each point in this range represents a unique volume when substituted into the formula.

Thinking about temperature ranges helps in understanding scenarios where temperature conditions are given, allowing specific volume predictions.

Algebraic Substitution

Algebraic substitution is a method used to find specific values of variables. In our exercise, substitution means inserting specific temperature values into the equation \( V = 20T \) to find the corresponding volumes. First, substitute 80°C, which gives \( V = 20 \times 80 = 1600 \) cc. Then, substitute 120°C, resulting in \( V = 20 \times 120 = 2400 \) cc.

• Allows us to find precise outcomes by inserting known values.
• Makes it possible to calculate outcomes within defined ranges like the 80°C to 120°C range.

This method is efficient because it reduces complexity and allows for quick calculations, especially with linear equations. Knowing how to substitute values is fundamental in solving problems where equations link different quantities, making it a key skill in mathematics and science.