Problem 4 - Algebra 2

Question

When a person swims underwater, the pressure in his or her ears varies directly with the depth at which he or she is swimming. Write a direct variation equation that represents this situation.

Step-by-Step Solution

Answer

The direct variation equation is \( P = kd \).

Step 1: Understanding Direct Variation

In a direct variation, when one variable increases, the other variable increases in such a way that their ratio remains constant. Here, the pressure variation is directly related to the depth of the water, so the pressure increases as the depth increases.

Step 2: Identifying Variables

Let's denote the pressure as \( P \) and the depth as \( d \). Since these are related by direct variation, we need to express this relation in the form of \( P = kd \) where \( k \) is the constant of variation.

Step 3: Writing the Direct Variation Equation

Using the relationship from direct variation, we can write the equation as \( P = kd \), which clearly shows that pressure \( P \) varies directly with depth \( d \). The constant \( k \), which is the constant of proportionality, depends on the specific conditions of the situation.

Key Concepts

Pressure and Depth Relation

Understanding the link between pressure and depth is crucial in many scientific and practical contexts. When you swim underwater, you might notice your ears feeling different. This is because as you go deeper, the water above you increases. This higher amount of water means more weight is pushing down, leading to greater pressure. This direct relationship suggests that pressure increases proportionately with depth.

Simply put, the deeper you swim, the more pressure is exerted on you by the water. This happens because the weight of the water above increases, squeezing the space around you. Knowing this relationship helps us anticipate changes, especially pertinent for divers or scientists studying oceanic environments. Recognizing that pressure and depth are directly linked helps in predicting and calculating these pressures accurately.

Variable Identification

When dealing with any equation, especially those involving direct variation, identifying the variables plays a key role. In such scenarios, you first need to decide which quantities are changing in relation to each other. For our underwater example:- The **pressure** exerted on the swimmer is one variable.- The **depth** of water the swimmer is at is the other variable.

By naming these variables, typically pressure as \( P \) and depth as \( d \), the process of understanding how they interact becomes clearer. These variables represent the quantities we are comparing against each other. It's crucial to identify them correctly to establish their mathematical relationship. Once you've pinpointed them, you can begin constructing the equation that describes their direct variation.

Writing Equations in Mathematics

Equations are foundational in mathematics, providing a way to express relationships clearly and succinctly. When dealing with direct variation, writing the equation correctly ensures that it can be easily interpreted and applied. In the context of our underwater example:- Since pressure \( P \) varies directly with depth \( d \), we express this with the equation \( P = kd \).- Here, \( k \) stands for the constant of variation, or how strongly pressure increases with depth.

Equations like this help simplify complex concepts into understandable formats. They enable us to predict outcomes — in this case, the pressure at a given depth — given the right conditions. By stating the equation, we make it easy for anyone to calculate changes in pressure based on changes in depth. Understanding how to construct and interpret such equations is an essential skill in math and science.