Problem 39 - 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. In the original equation, is \(P\) a function of \(d\) or is \(d\) a function of \(P ?\) Explain.

Step-by-Step Solution

Answer

The pressure \(P\) is a function of the depth \(d\), because the value of \(P\) depends on and changes with the value of \(d\).

Step 1: Understand the function

A function relates an output to one or more inputs. It’s like a machine where you input something, a process happens, and then you get different output. In the context of this task, if we input a certain depth (d) we get the pressure (P). So, we can say that P is a function of d.

Step 2: Apply to the formula

Applying this to the formula \(P = 64d + 2112\), we see that we put a value for \(d\) (depth) in, and what we get out is \(P\) (pressure). So, this shows that pressure \(P\) is a function of the depth \(d\).

Key Concepts

Defining Functions

In Algebra, a function is essentially a relationship between two variables, commonly represented as 'input' and 'output'. Imagine it as a special kind of rule that provides an output for every input. Functions come in various types and complexities, but at their core, they all serve to map inputs to outputs in a consistent and predictable manner.

For example, in the provided exercise, a scuba diver's pressure at a depth of \(d\) feet is described by the function \(P(d) = 64d + 2112\), where \(P\) stands for the total pressure exerted on the diver as depth changes. Here, depth \(d\) is the independent variable (input), and pressure \(P\) is the dependent variable (output).

This function tells us the pressure felt by a diver at any given depth. It denotes a clear method to calculate pressure based on depth. The rule here is simple: multiply the depth by 64 and then add 2112. This process defines the function and assures that for each depth value provided as input, we yield a unique pressure as output.

Functional Relationships

Functional relationships, in algebra, express how one quantity is related to another. A function demonstrates this relationship by pairing every input with exactly one output. In non-technical terms, it means that for every action, there is a specific and single reaction according to the function’s rule.

In the given exercise, we see the functional relationship between the depth \(d\) a diver reaches and the pressure \(P\) they experience. If we consider different depths such as 10 feet, 20 feet, or 50 feet, we can calculate the respective pressures using the function, and each depth will correspond to one particular pressure value.

Identifying Functions

How can one tell if a relationship is functional? A key identifier of a functional relationship is consistency - one input should not give multiple outputs. For instance, if we have the same depth \(d\), we should not calculate two different pressures; there's only one correct value of \(P\) for a given \(d\) in our function.

Linear Function

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. Its standard form is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) represents the y-intercept, the point where the line crosses the y-axis. These functions are among the simplest and most frequently encountered relationships in algebra.

In our scuba diving example, the function for pressure, \(P(d) = 64d + 2112\), is linear. The number 64 represents the slope \(m\), which indicates how quickly pressure increases per foot of depth, while 2112 is the y-intercept \(b\), the pressure at sea level or when \(d\) equals zero.

Straight-Line Relationships

The beauty of linear functions lies in their predictability and uniformity. For each additional foot of depth that a diver descends, the pressure increases by a constant rate of 64 pounds per square foot. This direct proportionality means that the graph of this function would be a straight line, rising to the right if graphed on a depth versus pressure coordinate system.

The concept of a linear function extends well beyond this example and is a foundational element for understanding more complex mathematical models in algebra and beyond.