Problem 73 - Algebra and Trigonometry

Question

Pressure and Depth At the surface of the ocean the water pressure is the same as the air pressure above the water, 15 \(\mathrm{lb} / \mathrm{in}^{2}\). Below the surface the water pressure increases by 4.34 \(\mathrm{lb} / \mathrm{in}^{2}\) for every 10 \(\mathrm{ft}\) of descent. (a) Find an equation for the relationship between pressure and depth below the ocean surface. (b) Sketch a graph of this linear equation. (c) What do the slope and \(y\) -intercept of the graph represent? Yd) At what depth is the pressure 100 Ib/in \(^{2} ?\)

Step-by-Step Solution

Answer

The pressure relationship with depth is linear: \( P = 0.434d + 15 \). At 196 feet, pressure is 100 lb/in².

Step 1: Understand the Initial Conditions

The initial pressure at the ocean's surface is given as 15 lb/in². We will call this initial condition \( P_0 = 15 \) lb/in². This represents the starting pressure at a depth of 0 feet.

Step 2: Establish the Rate of Pressure Increase

The pressure increases by 4.34 lb/in² for every 10 feet of descent. We can express this rate as a pressure increase per foot: \( \frac{4.34 \, \text{lb/in}^2}{10 \, \text{ft}} = 0.434 \, \text{lb/in}^2/\text{ft} \). This is our constant rate of change, or slope (\( m \)) of the linear equation.

Step 3: Write the Linear Equation

The pressure \( P \) at any depth \( d \) can be calculated using the slope-intercept form \( P = m \, d + P_0 \), where \( m = 0.434 \) lb/in²/ft and \( P_0 = 15 \) lb/in². Substituting these values, the equation becomes \( P = 0.434d + 15 \).

Step 4: Graph the Equation

We graph the equation \( P = 0.434d + 15 \), with the depth \( d \) on the x-axis and the pressure \( P \) on the y-axis. The y-intercept is 15 lb/in², which occurs when depth \( d = 0 \). The slope of the line is 0.434 lb/in²/ft, showing that pressure increases with depth.

Step 5: Interpret the Slope and Y-Intercept

The slope, 0.434 lb/in²/ft, represents the increase in pressure for each foot descended. The y-intercept, 15 lb/in², is the atmospheric pressure at the ocean's surface.

Step 6: Solve for Pressure of 100 lb/in²

To find the depth \( d \) where the pressure is 100 lb/in², we set \( P = 100 \) in the equation \( P = 0.434d + 15 \) and solve for \( d \): Substitute \( P = 100 \):\[ 100 = 0.434d + 15 \]Subtract 15 from both sides:\[ 85 = 0.434d \]Divide by 0.434:\[ d \approx 196 \] ft. Therefore, at a depth of about 196 feet, the pressure is 100 lb/in².

Key Concepts

Linear Equation

A linear equation is an essential mathematical concept indicating a straight-line relationship between two variables. In the context of the problem, the variables are pressure and depth. The nature of linear equations is that they show how one variable changes with respect to another.
For example, in our ocean scenario:

• Pressure increases as depth increases, adhering to a constant rate of change.
• The relationship is predictable, following a straight line when graphed.

Linear equations are generally represented in the form \( y = mx + c \), where \( y \) and \( x \) are variables, \( m \) is the slope, and \( c \) is the y-intercept. In our case, pressure \( P \) is a function of depth \( d \), thus our equation appears as \( P = 0.434d + 15 \).
Understanding linear equations helps us model real-world phenomena like how pressure builds up in water as you dive deeper.

Slope-Intercept Form

The slope-intercept form is a popularly used mathematical representation for linear equations, shown as \( y = mx + c \). This form makes it easy to derive and understand two important aspects: the slope (\( m \)) and the y-intercept (\( c \)).
In the context of pressure and depth:

• The slope \( m = 0.434 \) lb/in²/ft represents the rate at which pressure increases for each foot you descend. It shows the linear change of pressure with depth.
• The y-intercept \( c = 15 \) lb/in² denotes the pressure at the ocean's surface, which is the atmospheric pressure before diving below the water level.

This format is user-friendly for graphing and interpreting the relationship between pressure and depth. It provides a straightforward way to see how changes in depth influence pressure.

Graphing Concepts

Graphing allows us to visualize the relationship between variables like pressure and depth intuitively. When we graph the equation \( P = 0.434d + 15 \):

• The x-axis represents depth \( d \), and the y-axis represents pressure \( P \).
• The graph is a straight line due to the linear relationship indicated by the equation.
• The y-intercept is 15 lb/in², marking the starting point of the line when depth is zero.
• The slope of 0.434 lb/in²/ft is reflected in the angle of the line, showcasing how pressure steadily increases with greater depths.

Graphing these relations provides a clear, visual aid to understanding how each foot of descent impacts pressure. It's a vital skill for comprehending changes and trends within data.

Rate of Change

The rate of change is a critical concept in understanding how one quantity affects another. It is vividly illustrated in this exercise as the amount pressure increases with each foot descended in the ocean. In mathematical terms, this concept is expressed as the slope in a linear equation.
Here, the rate of change is:

• 0.434 lb/in²/ft, indicating how much pressure increases per foot of depth.
• This consistency allows us to predict the pressure at any given depth.

In practical applications, knowing the rate of change provides insights into expected outcomes and helps in planning dives, equipment needs, or measuring the relative safety of descending to specific depths. This precise understanding of pressure's rate of change with depth is crucial for activities such as diving and underwater exploration.