Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Nuclear experiments performed in the ocean vaporize large quantities of salt water. Salt boils and turns into vapor at \(1738 \mathrm{~K}\). After being vaporized by a 10 -megaton force, the salt takes at least \(8-10\) seconds to cool enough to crystallize. The amount of salt \(A\) that has crystallized \(t\) seconds after an experiment is sometimes calculated using \(A=k \sqrt{t / T}\), where \(k\) and \(T\) are constants. Solve this equation for \(t\).

Step-by-Step Solution

Answer

\( t = T \left( \frac{A}{k} \right)^2 \)

Step 1: Understand the Formula

The formula given in the problem is \( A = k \sqrt{\frac{t}{T}} \). We need to solve this equation for the variable \( t \), which means we want to express \( t \) in terms of \( A \), \( k \), and \( T \).

Step 2: Isolate the Square Root

To isolate the square root, divide both sides of the equation by \( k \), resulting in \( \frac{A}{k} = \sqrt{\frac{t}{T}} \).

Step 3: Square Both Sides

To remove the square root, square both sides of the equation, giving \( \left( \frac{A}{k} \right)^2 = \frac{t}{T} \).

Step 4: Solve for t

Multiply both sides by \( T \) to solve for \( t \), leading to \( t = T \left( \frac{A}{k} \right)^2 \).

Key Concepts

Crystallization

Crystallization is a fascinating process that occurs when a solute turns from a liquid solution into a solid, forming a crystal structure. This process happens due to the decrease in solubility when the solution cools or when it undergoes a physical change. In the context of nuclear experiments, when ocean water is vaporized, the crystallization of salt is inevitable once it cools down.
Understanding crystallization is crucial for several fields:

• In our everyday life, such as in the formation of snowflakes or salt from seawater.
• Industrial applications, like the purification of products via crystallization.
• Scientific research, where crystallization helps understand molecular structures.

In essence, crystallization involves two main steps: nucleation, where molecules gather to form a stable cluster, and growth, where additional molecules stick to the structure, forming a larger crystal.

Nuclear Experiments

Nuclear experiments in aquatic environments, although highly impactful, help to understand the behavior and effects of nuclear reactions. These experiments often involve detonation tests, which can drastically alter the immediate surroundings, causing fractures in rock formations and vaporizing large quantities of water.
Nuclear testing in oceans has led to significant questions regarding:

• Environmental impacts, including marine life disruption.
• The chemical reactions performed under extreme conditions, such as salt crystallization.
• Long-term effects on oceanic ecosystems and geology.

Such experiments, albeit controversial, provide essential scientific data that drive policy making and safety regulations.

Mathematical Modeling

Mathematical modeling is a process used to represent real-world phenomena through equations and mathematical expressions. In the problem given, the equation \( A = k \sqrt{\frac{t}{T}} \) models the crystallization of salt over time. It defines how various factors influence the amount of salt crystallized after vaporization.
Key aspects of mathematical modeling include:

• Using mathematical expressions to simulate physical processes.
• Making predictions and understanding complex systems.
• Providing insights that aren't easily observed through experiments alone.

By rearranging the equation to solve for \( t \), students learn how to manipulate models to find unknown variables based on known quantities.

Square Root Isolation

Square root isolation is a mathematical technique used to solve equations involving square roots. It requires manipulating the equation so that the term with the square root stands alone on one side. This process is essential when solving equations like \( A = k \sqrt{\frac{t}{T}} \), where we need to find \( t \).
Steps to isolate a square root include:

• First, divide both sides by any coefficient in front of the square root. In our exercise, this is \( k \), resulting in \( \frac{A}{k} = \sqrt{\frac{t}{T}} \).
• Second, square both sides to remove the square root, leading to \( \left( \frac{A}{k} \right)^2 = \frac{t}{T} \).
• Finally, solve for the remaining terms to find the unknown variable.

This method is fundamental in algebra, helping to simplify and solve various mathematical problems.