Problem 103 - College Algebra and Calculus: An Applied Approach

Question

The pressure (in atmospheres) exerted on a scuba diver's body has a linear relationship with the diver's depth. At sea level (or a depth of 0 feet), the pressure exerted on a diver is 1 atmosphere. At a depth of 99 feet, the pressure exerted on a diver is 4 atmospheres. Write a linear equation to describe the pressure \(p\) (in atmospheres) in terms of the depth \(d\) (in feet) below the surface of the sea. What is the rate of change of pressure with respect to depth? (Source: PADI Open Water Diver Manual)

Step-by-Step Solution

Answer

The linear equation describing the pressure \(p\) in terms of the depth \(d\) below the surface of the sea is \(p = 0.0303d + 1\). The rate of change of pressure with respect to depth is 0.0303 atmospheres per foot.

Step 1: Understand the linear equation form

A linear equation in two variables is typically written in the form \(y = mx + c\), where \(m\) is the slope of the line, which represents the rate of change, and \(c\) is the y-intercept, which is the point at which the line crosses the y-axis. Here, \(y\) is the pressure \(p\), \(x\) is the depth \(d\), \(m\) is the rate of change, and \(c\) will be the pressure at the sea level.

Step 2: Calculate the slope

The slope of a line, which represents the rate of change, is calculated from two points on the line as \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \). Substituting the given points, (0,1) and (99,4) for pressure p (y) and depth d (x) respectively, into the formula to calculate the slope, we get: \( m = \frac{4 - 1}{99 - 0} = \frac{3}{99} = 0.0303 \).

Step 3: Substitute the slope and the y-intercept

Now that the slope \(m = 0.0303\) and the y-intercept (which is the pressure at sea level, or 1 atmosphere) have been found, substitute into the equation \(p = md + c\). Substituting \(m = 0.0303\) and \(c = 1\) gives: \(p = 0.0303*d + 1\)

Key Concepts

Slope

The slope in a linear equation is a very important concept when dealing with two variables like pressure and depth. When you see an equation in the form of \( y = mx + c \), the \( m \) stands for the slope. It tells us how much \( y \) changes for each unit increase in \( x \). In simpler terms, it's the steepness of the line.
For the scuba diver, we looked at how the pressure changes as the diver goes deeper. We used two points: at sea level (0 feet, 1 atmosphere) and at 99 feet deep (4 atmospheres). The slope \( m \) is found by the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Substitute the numbers:

• \( m = \frac{4 - 1}{99 - 0} \)
• \( m = \frac{3}{99} \)
• \( m = 0.0303 \)

This slope, 0.0303, indicates that for every foot of descent, the pressure increases by approximately 0.0303 atmospheres.

Rate of Change

Rate of change is essentially another way of talking about the slope. It's all about understanding how one variable affects the other. In our example (with a scuba diver), the rate of change of pressure with respect to depth tells us how much the pressure increases as the diver goes deeper.
In practical terms, knowing the rate of change helps divers estimate how much pressure they'll experience at different depths without calculating it from scratch every time. It's very handy! For the given problem:

• The rate of change is 0.0303.
• This means increasing the depth by 1 foot increases the pressure by 0.0303 atmospheres.
• The rate of change remains constant because it's a linear relationship.

Understanding the rate of change aids in planning dives safely and effectively.

Y-intercept

The y-intercept in a linear equation is like a starting point. It's the value of \( y \) when \( x \) is zero, represented by \( c \) in the equation \( y = mx + c \). For divers, this point is crucial because it shows the pressure they start with, before they even begin to descend.
In the given exercise, the y-intercept is 1. This value tells us the pressure at sea level (depth of 0 feet). So, even when the diver hasn't dived at all, there is already a pressure of 1 atmosphere exerted on them. Here are some vital points:

• The y-intercept, \( c \), in our scenario is 1.
• This occurs at a depth of 0 feet, where there's no additional pressure from water.
• The y-intercept forms a baseline for measuring how pressure increases with depth.

By knowing the y-intercept, divers can better interpret how pressure changes as they change the depth of their dive.