Problem 3 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. An ocean diver models the pressure \(P\left(\text { in } \operatorname{lb} / \mathrm{in}^{2}\right)\) at depth \(d\) (in ft) by $$ P=0.45 d+14.7 $$ (a) Find the pressure \(P\) at a depth of 180 \(\mathrm{ft}\) . (b) Find the pressure \(P\) at a depth of 330 \(\mathrm{ft}\) .

Step-by-Step Solution

Answer

(a) 95.7 psi (b) 163.2 psi

Step 1: Understand the Model

The diver's model for pressure is given by the equation \( P = 0.45d + 14.7 \), where \( P \) is the pressure in pounds per square inch (psi) and \( d \) is the depth in feet. This model indicates that pressure increases linearly with depth, starting from a base pressure of 14.7 psi, which represents atmospheric pressure at the surface.

Step 2: Substitute Depth for Case (a)

To find the pressure at a depth of 180 feet, substitute \( d = 180 \) into the model equation. The formula becomes:\[ P = 0.45 \times 180 + 14.7 \]

Step 3: Calculate Pressure for Case (a)

Perform the multiplication first: \( 0.45 \times 180 = 81 \). Then, add the result to the atmospheric pressure: \( 81 + 14.7 = 95.7 \). Thus, the pressure at 180 feet is 95.7 psi.

Step 4: Substitute Depth for Case (b)

To find the pressure at a depth of 330 feet, substitute \( d = 330 \) into the model equation. It becomes:\[ P = 0.45 \times 330 + 14.7 \]

Step 5: Calculate Pressure for Case (b)

Multiply 0.45 by 330: \( 0.45 \times 330 = 148.5 \). Add this result to the atmospheric pressure: \( 148.5 + 14.7 = 163.2 \). Therefore, the pressure at 330 feet is 163.2 psi.

Key Concepts

Linear Functions

A linear function is a mathematical expression that models a constant rate of change between two variables. In this context, the linear function is used to describe how pressure changes with depth underwater. The equation for pressure, \( P = 0.45d + 14.7 \), illustrates this relationship.
Here, \( P \) represents the pressure at a certain depth \( d \). The coefficient \( 0.45 \) indicates the rate at which pressure increases with each foot of depth.
This means that for every additional foot underwater, the pressure goes up by 0.45 pounds per square inch (psi). The constant term \( 14.7 \) represents the atmospheric pressure at the water's surface.

• Constant rate of change: 0.45 psi per foot
• Starting value (y-intercept): 14.7 psi, atmospheric pressure

Understanding this equation helps in predicting how pressure varies at different depths, crucial for activities such as scuba diving and submarine engineering.

Depth Measurement

In underwater activities, depth measurement is crucial for ensuring safety and monitoring environmental conditions. Depth is simply the vertical distance from the water's surface to a point beneath it. In this model, depth \( d \) is measured in feet.
The linear function used by the diver relies on accurate depth measurements to predict pressure accurately, ensuring divers are aware of the conditions they face. This affects various underwater processes, such as:

• Monitoring pressure changes with depth
• Planning for safe diving practices
• Ensuring equipment can withstand different pressure levels

By substituting different values of depth into the model, divers can calculate the corresponding pressure and prepare appropriately for their dives. Thus, understanding how depth affects pressure is vital for safe underwater exploration.

Mathematical Modeling

Mathematical modeling is the process of using equations to represent real-world situations. In this exercise, the formula \( P = 0.45d + 14.7 \) is a mathematical model that represents the relationship between underwater pressure and depth.
Models like this simplify complex systems, allowing for predictions and analyses. When creating a model, it's essential to:

• Identify variables of interest (e.g., pressure \( P \) and depth \( d \))
• Determine the type of relationship (linear, in this case)
• Ensure the accuracy of constants (like the base pressure of 14.7 psi)

Such models are valuable because they provide a basis for predictions and decision-making in engineering, physics, and environmental studies. By using mathematical models, we can better understand and interact with the world around us.