Problem 69 - Applied Calculus

Question

GENERAL: Water Pressure At a depth of \(d\) feet underwater, the water pressure is \(p(d)=0.45 d+15\) pounds per square inch. Find the pressure at: a. The bottom of a 6 -foot-deep swimming pool. b. The maximum ocean depth of 35,000 feet.

Step-by-Step Solution

Answer

17.7 psi at 6 feet and 15765 psi at 35,000 feet.

Step 1: Understand the Function

The pressure function is given as \( p(d) = 0.45d + 15 \), where \( d \) is the depth in feet. This means the pressure increases by 0.45 pounds per square inch (psi) for each foot of depth, starting at a base pressure of 15 psi.

Step 2: Substitute the Depth for Scenario (a)

For (a), the depth \( d \) is 6 feet. Substitute \( d = 6 \) into the function: \( p(6) = 0.45 \times 6 + 15 \).

Step 3: Calculate Pressure for Scenario (a)

Calculate \( 0.45 \times 6 = 2.7 \), then add 15: \( p(6) = 2.7 + 15 = 17.7 \) psi. This is the pressure at the bottom of the 6-foot pool.

Step 4: Substitute the Depth for Scenario (b)

For (b), the depth \( d \) is 35,000 feet. Substitute \( d = 35000 \) into the function: \( p(35000) = 0.45 \times 35000 + 15 \).

Step 5: Calculate Pressure for Scenario (b)

Calculate \( 0.45 \times 35000 = 15750 \), then add 15: \( p(35000) = 15750 + 15 = 15765 \) psi. This is the pressure at the maximum ocean depth.

Key Concepts

Applied Calculus

Applied Calculus is a practical branch of mathematics that involves using calculus concepts to solve real-world problems. This field is broad, allowing us to deal with dynamic systems and changes, like calculating pressure at varying depths underwater.
In this exercise, we use a linear model represented by the pressure function to determine how pressure changes with depth. Applied calculus empowers us to perform this calculation by understanding how rates of change work. Through simple mathematical operations and models, we can solve complex problems in everyday life.
Understanding the basics of calculus, like derivatives and integrals, helps understand changes and movement in physical systems, such as pressure increase with depth. Although we only use basic arithmetic here, the underlying concepts of calculus are ever-present as we examine how pressure scales with depth.

Pressure Function

A pressure function is a mathematical model that represents how pressure changes in accordance with different variables, such as depth. In this exercise, the pressure function is given as:
\[ p(d) = 0.45d + 15 \] where:

• \( p(d) \): pressure in pounds per square inch (psi),
• \( d \): depth in feet.

Each foot of depth increases pressure by 0.45 psi, starting from an initial pressure of 15 psi.
The initial pressure accounts for standard atmospheric pressure that water exerts at the surface. Mathematical functions like this help us predict and understand how variables such as pressure react to different influences, making them vital tools in scientific calculations.

Depth Calculation

Depth calculation involves finding the water pressure at different depths with the help of a pressure function. By substituting specific depth values into the function, we can determine the pressure at that depth.
In the original exercise, this involved calculating the pressure at a 6-foot swimming pool depth and a maximum ocean depth of 35,000 feet.
The process involved:

• Substituting \(d = 6\) into the pressure function to find a pool water pressure of 17.7 psi.
• Substituting \(d = 35000\) to find the ocean pressure would reach 15,765 psi.

This simple substitution and arithmetic give us precise pressure values at varying depths, showcasing the real-life applications of mathematical functions.

Mathematical Modeling

Mathematical modeling refers to the process of using mathematics to represent a system or phenomenon in the real world. This exercise is a typical example, using a mathematical equation to model how pressure increases with depth underwater.
Mathematical models enable us to simulate conditions, forecast outcomes, and analyze systems' behavior in different scenarios. In this context, the pressure function \[ p(d) = 0.45d + 15 \]represents the relationship between depth and pressure. Models like these are invaluable as they allow researchers and engineers to predict pressures at various underwater depths when planning constructions or studies on marine life.
Through these models, complex physical realities become accessible through calculations, providing insights that drive innovation and safety in numerous fields.