Problem 69 - Algebra 2

Question

The bottom of the Mariana Trench in the Pacific Ocean is 6.8 miles below sea level. Water pressure in the ocean is represented by the function \(f(x)=1.15 x\) , where \(x\) is the depth in miles and \(f(x)\) is the pressure in tons per square inch. Find the pressure in the Mariana Trench.

Step-by-Step Solution

Answer

The pressure at the bottom of the Mariana Trench is 7.82 tons per square inch.

Step 1: Identify the given information

We are given the depth of the Mariana Trench as 6.8 miles below sea level and a function that describes the water pressure at a given depth: \(f(x)=1.15x\).

Step 2: Substitute the known value into the function

To find the pressure at the bottom of the trench, substitute \(x = 6.8\) into the function \(f(x) = 1.15x\). This gives \(f(6.8) = 1.15 \times 6.8\).

Step 3: Calculate the pressure

Perform the multiplication to calculate the pressure: \(f(6.8) = 1.15 \times 6.8 = 7.82\).

Step 4: Interpret the result

The pressure at a depth of 6.8 miles in the Mariana Trench is 7.82 tons per square inch, according to the function provided.

Key Concepts

Mariana Trench

The Mariana Trench is the deepest part of the world's oceans. It is located in the western Pacific Ocean, and its deepest point, known as the Challenger Deep, reaches depths of about 6.8 miles (or approximately 11 kilometers) below sea level. Being the lowest known point on Earth's surface, it presents unique challenges and discoveries for scientists and explorers. The extreme conditions at such depths, including high water pressure and pitch darkness, make the trench a fascinating subject of study. It serves as a crucial natural laboratory for understanding the Earth's geology, marine biology, and physics, among other fields.

Ocean Depth

Ocean depth plays a critical role in determining many environmental and physical properties of the marine environment. At great depths, such as those found in the Mariana Trench, several factors are in play:

• Light: Light diminishes rapidly with depth, leaving the deepest parts in complete darkness.
• Temperature: Water temperature decreases as depth increases, often reaching near freezing at the greatest depths.
• Pressure: The pressure increases significantly, with each additional mile of depth resulting in increased force on surfaces.

Understanding ocean depth is essential for calculating pressures that affect marine vessel construction and the behavior of sound and light underwater. This knowledge helps design submersible vehicles, and it is vital for telecommunications that utilize underwater cables.

Function Evaluation

Function evaluation involves substituting a given value for the variable in a function to find the result. In our exercise, we have a linear function representing water pressure in the ocean: \[ f(x) = 1.15x \]Here, \(x\) is the ocean depth in miles, and \(f(x)\) represents the pressure in tons per square inch. To evaluate the function, you replace \(x\) with the specific depth you are analyzing. For instance, substituting \(x = 6.8\) miles yields:\[ f(6.8) = 1.15 \times 6.8 = 7.82 \]This value means that the pressure at the deepest part of the Mariana Trench is 7.82 tons per square inch. This process helps us understand how pressure increases proportionally with depth, guided by the linear nature of the function.

Pressure Measurement Concepts

Pressure in a fluid, such as seawater, is the force exerted perpendicular to a surface area. High-pressure measurements are essential in exploring oceanic features like the Mariana Trench. With increased depth, the weight of the water column above exerts greater force, calculated as pressure. The concept of pressure is fundamental in various scientific and engineering applications. It is essential for designing equipment and structures that can withstand significant forces. In our exercise, the pressure was determined using a linear function, hinting at a consistent increase relative to depth. The unit used here is tons per square inch, a measure of force per unit area, making it easier to equip subsea structures and flooding scenarios with the necessary strength to handle deep-sea pressures effectively.