Problem 23 - Algebra Form and Function

Question

Let \(f(T)\) be the volume in liters of a balloon at temperature \(T^{\circ} \mathrm{C}\). If \(f(40)=3\) (a) What are the units of the 40 and the 3 ? (b) What is the volume of the balloon at \(40^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

Answer: The units of 40 are degrees Celsius, the units of 3 are liters, and the volume of the balloon at 40 degrees Celsius is 3 liters.

Step 1: Identify the units of 40 and 3

The exercise mentions that \(T\) is in degrees Celsius and \(f(T)\) is the volume in liters. So, 40 represents the temperature \(T\) in degrees Celsius, and 3 represents the volume in liters.

Step 2: Find the volume of the balloon at 40 degrees Celsius

We are given that \(f(40) = 3\), which means the volume of the balloon at a temperature of \(40^{\circ} \mathrm{C}\) is 3 liters. Therefore, the volume of the balloon at \(40^{\circ} \mathrm{C}\) is 3 liters. (a) The units of 40 are degrees Celsius (\(^{\circ}\mathrm{C}\)), and the units of 3 are liters. (b) The volume of the balloon at \(40^{\circ}\mathrm{C}\) is 3 liters.

Key Concepts

Function

In algebra, a function is a special relationship that maps each input from a set of inputs to exactly one output in a set of outputs. A function describes how each element of one set relates to exactly one element of another set. For example, if you have a function \(f(T)\) in this exercise, \(T\) represents the input (temperature) and \(f(T)\) is the output (volume of the balloon).
The notation \(f(T)\) signifies that \(f\) is a function that is dependent on \(T\).
This relationship is crucial as it tells us how the volume of the balloon can vary with temperature.

• An input like \(T = 40\) leads to an output \(f(40) = 3\), showing that at 40°C, the volume is 3 liters.
• This one-to-one correspondence defines the core property of a function.

Understanding functions is key to solving problems that model relationships in the real world, such as predicting changes in balloon volume based on temperature.

Temperature

Temperature is a measure of how hot or cold something is and is an essential variable in many scientific formulas and functions. It is commonly measured in degrees Celsius (°C) in most parts of the world. In this context, temperature determines the physical characteristics of individual systems, like the volume of a gas in a balloon.
As temperature changes, many physical properties can change as well:

• When temperature increases, gases generally expand, causing an increase in volume.
• Conversely, a decrease in temperature might cause gases to contract.

In algebraic functions like the one in this exercise, temperature plays a vital role as an independent variable that influences the function’s outcomes, such as how the balloon's volume alters with temperature.

Units of Measurement

Units of measurement provide a standard for expressing and comparing the quantity of physical quantities. In mathematical functions and algebra, correctly understanding and applying these units is crucial. Here, we deal with two types of units:

• Temperature Units (Degrees Celsius): Represented by °C, this unit measures the hotness or coldness of the environment affecting objects like balloons. It’s the worldwide standard in most scientific equations, except for a few regions where Fahrenheit is used.
• Volume Units (Liters): This unit measures how much space an object or substance occupies, in this case, the balloon's volume. Liters are a common metric for measuring liquid capacities.

Correctly interpreting units ensures accurate communication and understanding when solving real-world problems. For practical situations, matching the right unit with its quantity is critical to avoiding misunderstandings or errors, such as confusing temperature and volume.