Problem 38 - Algebra 1

Question

Use the following information. If a scuba diver starts at sea level, the pressure on the diver at a depth of \(d\) feet is given by the formula \(P=64 d+2112,\) where \(P\) represents the total pressure in pounds per square foot. Suppose the current pressure on a diver is 4032 pounds per square foot. What is the diver's current depth?

Step-by-Step Solution

Answer

The diver's current depth is 30 feet.

Step 1: Understand the Problem Equation

We are given the equation \(P = 64d + 2112\), where \(P\) represents diver pressure and \(d\) is the given depth.

Step 2: Rearrange the Equation for 'd'

In order to figure out the depth given the pressure, it is useful to rewrite this fairly straightforward equation in terms of \(d\). By subtracting \(2112\) from both sides and subsequently dividing by \(64\), we get: \(d = (P - 2112) / 64\).

Step 3: Substitute the Given Pressure into the Equation

Now the given pressure of \(4032\) pounds per square foot can be substituted into the equation: \(d = (4032-2112) / 64\).

Step 4: Solve the Equation

Finish the calculation to obtain depth \(d = 1920 / 64 = 30\) feet.

Key Concepts

Linear Equations

Linear equations are fundamental tools in algebra, often used to model relationships between variables. They express a relationship in which one variable depends on another in a linear manner, meaning the graph of this equation is a straight line. A linear equation can be identified by its standard form, which is usually represented as \( y = mx + b \) where:

• \( y \) is the dependent variable.
• \( m \) represents the slope of the line.
• \( x \) is the independent variable.
• \( b \) is the y-intercept.

The key to solving a linear equation is to isolate the variable of interest by performing equivalent transformations. This involves operations like adding, subtracting, multiplying, or dividing both sides of the equation by the same number. In the scuba diving example, the linear equation \( P = 64d + 2112 \) relates pressure and depth, requiring rearrangement to make depth \( d \) the subject. This process involved subtracting 2112 from both sides and then dividing by 64, showcasing classic maneuvers applied in solving linear equations.

Depth Calculation

Depth calculation is crucial in scenarios such as diving, where knowing how deep one is below the surface can impact safety and performance. In many exercises like this, the goal is to find the depth based on given parameters using an established formula.

In our case, the formula provided is \( d = \frac{P - 2112}{64} \). Given a pressure \( P \) of 4032 pounds per square foot, we substitute this value into the equation to find:

• Subtracting 2112 from \( 4032 \) gives \( 1920 \).
• Dividing \( 1920 \) by 64 will provide the depth \( d \).

The algebra thus simplifies to \( d = 30 \), representing the diver's depth of 30 feet. Such calculations highlight the importance of understanding and manipulating the relationship between different elements, viscerally dug out through careful arithmetic.

Pressure Formula

The pressure formula in physics often serves as a gateway to understanding different environmental and physical pressures affecting matter. In depth and diving-related problems, pressure is a function of several variables, including depth. The formula \( P = 64d + 2112 \) serves as a model for calculating the pressure experienced by a diver at a certain depth underwater.

This specific equation can be broken down as follows:

• The term \( 64d \) represents how pressure increases with depth \( d \). The coefficient 64 is a constant that indicates how much pressure increases per foot of depth.
• The constant \( 2112 \) stands for the initial pressure at the surface, which includes atmospheric pressure exerted at sea level.

Understanding this relationship is integral for calculations involving depth and pressure. The formula allows divers to determine the increase in pressure they will experience at various depths, an essential factor in planning safe dives. Knowing and applying the pressure formula is fundamental in many physics-based real-world applications, particularly in fields involving fluid dynamics.