Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Salinity of the ocean refers to the amount of dissolved material found in a sample of seawater. Salinity \(S\) can be estimated from the amount \(C\) of chlorine in seawater using \(S=0.03+1.805 C\), where \(S\) and \(C\) are measured by weight in parts per thousand. Approximate \(C\) if \(S\) is \(0.35\).

Step-by-Step Solution

Answer

The approximate amount of chlorine, \( C \), is 0.177 parts per thousand.

Step 1: Understanding the Problem

We need to find the value of chlorine content \( C \) given the salinity \( S \) of the ocean. The relationship between salinity \( S \) and chlorine \( C \) is given by the equation \( S = 0.03 + 1.805C \). In our case, the salinity \( S \) is 0.35 parts per thousand.

Step 2: Rearrange the Equation

To find \( C \), we need to rearrange the equation \( S = 0.03 + 1.805C \) to solve for \( C \). This can be done by isolating \( C \) on one side of the equation: \( C = \frac{S - 0.03}{1.805} \).

Step 3: Substitute the Given Value

We substitute the given value of salinity \( S = 0.35 \) into the rearranged equation: \[ C = \frac{0.35 - 0.03}{1.805} \].

Step 4: Calculate the Value of C

Now, perform the arithmetic operations: first, subtract 0.03 from 0.35 to get 0.32, and then divide by 1.805. This gives \( C \approx 0.1773 \).

Key Concepts

Chlorine Content Calculation

To determine the chlorine content in seawater, we use a mathematical relationship between salinity, denoted as \( S \), and chlorine content, \( C \). This relationship is given by the equation:\[ S = 0.03 + 1.805C \]The task is to find \( C \) when \( S \) is known. First, the equation must be rearranged to solve for \( C \):\[ C = \frac{S - 0.03}{1.805} \]In this exercise, with a given salinity \( S = 0.35 \), simply plug this value into the equation:\[ C = \frac{0.35 - 0.03}{1.805} \]By performing these calculations, you find that \( C \approx 0.1773 \) parts per thousand. This value represents the amount of chlorine in the seawater.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when graphed. They typically take the form:\[ y = mx + b \]In the context of salinity estimation, the equation \( S = 0.03 + 1.805C \) is a specific linear equation where:

• \( S \) is the dependent variable (salinity).
• \( C \) is the independent variable (chlorine content).
• \( 0.03 \) is the y-intercept, representing a base salinity when \( C \) is zero.
• \( 1.805 \) is the slope, indicating the rate of change in salinity per unit of chlorine.

When presented with a linear equation like this, solving for one variable involves rearranging the equation, as demonstrated in the chlorine content calculation section.

Arithmetic Operations

Arithmetic operations are fundamental in mathematics, involving basic procedures like addition, subtraction, multiplication, and division.In this problem, we calculate chlorine content using:

• Subtraction: to modify the base salinity from the measured value, \( 0.35 - 0.03 = 0.32 \).
• Division: to scale the difference by the slope (1.805), resulting in \( \frac{0.32}{1.805} \).

Performing these operations, arithmetic correctly determines the chlorine content, demonstrating their central role in solving equations effectively. Consistent practice with these skills enhances accuracy and speed in computations.

Chemical Properties of Seawater

Seawater is rich in dissolved substances like salts, minerals, and gases, contributing to its salinity. Salinity is a measure of these dissolved materials and is crucial for understanding ocean chemistry and biology. Chlorine, along with sodium, is a primary contributor to seawater's salinity. By calculating chlorine content, scientists can infer total salinity levels and make important ecological and industrial applications.

• Salinity influences seawater density, impacting ocean currents and climate.
• It plays a crucial role in marine life survival, affecting buoyancy and metabolic processes.
• Studying salinity aids in understanding the ocean's role in the Earth's water cycle and climate regulation.

Estimating salinity through chlorine content provides a practical and reliable method for evaluating oceanic and environmental conditions.