Problem 117 - A Graphical Approach to College Algebra

Question

surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$ f(x)=31.5+1.1 \log (x+1) $$ models salinity to depths of 1000 meters at a latitude of \(57.5^{\circ} \mathrm{N} .\) The variable \(x\) is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D\), Global Physical Climatology, Academic Press.) Approximate analytically the depth where the salinity equals 33

Step-by-Step Solution

Answer

The depth where salinity equals 33 is approximately 22.13 meters.

Step 1: Understand the Function

The function \( f(x) = 31.5 + 1.1 \log(x + 1) \) represents the salinity in grams of salt per kilogram of seawater at a depth \( x \) in meters.

Step 2: Set Up the Equation

We need to find the depth \( x \) where the salinity \( f(x) = 33 \). Set the function equal to 33: \[ 31.5 + 1.1 \log(x + 1) = 33 \]

Step 3: Isolate the Logarithmic Expression

Subtract 31.5 from both sides of the equation:\[ 1.1 \log(x + 1) = 1.5 \]

Step 4: Solve for the Logarithm

Divide both sides by 1.1 to isolate the logarithm:\[ \log(x + 1) = \frac{1.5}{1.1} \]

Step 5: Calculate the Logarithmic Value

Compute \( \frac{1.5}{1.1} \):\[ \log(x + 1) = 1.3636 \] (approximately)

Step 6: Convert from Logarithmic to Exponential Form

Recall that if \( \log_b(a) = c \), then \( a = b^c \). Here, \( \log(x + 1) \) is base 10, so:\[ x + 1 = 10^{1.3636} \]

Step 7: Calculate the Exponential Value

Calculate \( 10^{1.3636} \):\[ x + 1 \approx 23.13 \]

Step 8: Solve for x

Subtract 1 from both sides to find \( x \):\[ x \approx 22.13 \]

Step 9: Check the Approximation

If you want to verify the approximation, substitute \( x = 22.13 \) back into the function to see if it results in a salinity close to 33.

Key Concepts

Salinity Modeling

Salinity modeling is crucial in understanding the composition of seawater, which varies depending on factors like evaporation, rainfall, and geographical location. In the given function, the term \( f(x) = 31.5 + 1.1 \log(x + 1) \) provides a way to model salinity changes with depth. This model reflects real-world phenomena, where higher depths may see varying salinity levels due to different environmental conditions.

Interestingly, the function's components translate actual events into mathematical language:

• \(31.5\) represents a base level of salinity, seen as a constant.
• \(1.1 \log(x + 1)\) expresses variation with depth \(x\), utilizing a logarithmic relationship that reflects how salinity changes gradually rather than abruptly.

The use of logarithmic functions in salinity modeling helps capture the nonlinear nature of these changes, providing a smooth curve representing gradual transitions. This method allows researchers to predict salinity levels at various depths in oceanography studies, informing ecological studies and navigation safety.

Depth Calculation

Calculating depth in the context of salinity modeling involves identifying the depth \( x \) where a specific salinity level occurs. For instance, the task is to find \( x \) at which the salinity equals 33 grams per kilogram.

To find this, we begin by setting the function \( f(x) = 31.5 + 1.1 \log(x + 1) = 33 \). Then, it's necessary to isolate and solve for \( x \), using steps like:

• Subtract the constant base salinity, \(31.5\), from both sides to simplify the equation.
• Following the removal, divide by \(1.1\) to obtain the pure logarithm \(\log(x + 1)\).
• Convert the logarithmic equation into an exponential form to solve for \(x\).

Finally, by solving these steps, the depth where this particular salinity level is found appears at around \(22.13\) meters. Understanding this process is essential for anyone studying how certain measurements and equations are adjusted to reveal specific, real-world insights related to depth.

Mathematical Functions

Mathematical functions like those used in salinity modeling play a pivotal role in translating natural phenomena into a form that can be analyzed and interpreted. Here, the use of logarithmic functions is significant as it manages nonlinear relationships effectively.

Logarithms are particularly handy when dealing with data that increase or decrease exponentially. The function \( f(x) = 31.5 + 1.1 \log(x + 1) \) exemplifies how such mathematical expressions can model a real-case situation like ocean salinity variation. This is crafted so that:

• It smoothly incorporates small changes over large scales, explaining the non-linear increase in salinity with depth.
• The base of the logarithm, typically 10, provides easy conversion between logarithmic and exponential expressions, simplifying calculations.

Understanding these mathematical patterns is not only crucial to resolving specific salinity equations but also broadly applies across diverse scientific fields, providing tools to solve problems involving growth, decay, and change in various contexts.