Question

The approximate density of seawater at a depth of \(h\) mi is $$d=64.0 e^{0.00676 h}\left(\mathrm{lb} / \mathrm{ft}^{3}\right)$$ Find the depth \(h\) at which the density will be \(64.5 \mathrm{lb} / \mathrm{ft}^{3}.\)

Short answer

The depth h at which the density of seawater will be approximately 64.5 lb/ft³ is around 1.15 miles.

Step-by-step solution

- Set up the equation

To find the depth at which the density will be 64.5 lb/ft³, set up the equation using the density formula given: \(64.0e^{0.00676h} = 64.5\).

- Isolate the exponential term

We need to isolate the exponential term to solve for h. Divide both sides of the equation by 64.0 to get \(e^{0.00676h} = \frac{64.5}{64.0}\).

- Calculate the right side

Perform the division on the right side of the equation: \(e^{0.00676h} = 1.0078125\).

- Take natural logarithm of both sides

To solve for h, take the natural logarithm (ln) of both sides of the equation to get \(0.00676h = \ln(1.0078125)\).

- Calculate the natural logarithm

Find the natural logarithm of 1.0078125: \(\ln(1.0078125) \approx 0.007758\).

- Solve for h

Finally, divide both sides of the equation by 0.00676 to solve for h: \(h = \frac{0.007758}{0.00676} \approx 1.1479\).

Key concepts

Natural Logarithm

The natural logarithm, denoted as ln, is a mathematical function that's typically used to solve equations where the variable of interest is in the exponent of an expression—the case of exponential equations. Specifically, it's the inverse of the exponential function. When dealing with the natural logarithm, we're looking at the power to which the number e (an irrational and transcendental number approximately equal to 2.71828) must be raised to obtain a certain number. Let's say you have an equation like \(e^x = b\). To find the value of \(x\), we take the natural logarithm of both sides, which allows us to isolate the variable: \(x = \text{ln}(b)\).

The importance of this function in solving exponential equations cannot be overstated. During this operation, ln and e cancel each other out because they are inverse functions, leaving us with the argument of the exponential function, which simplifies the problem. Moreover, using ln is particularly helpful because it allows us to work with growth and decay rates, which are commonly expressed as exponential functions, and find constants or variables within these expressions. In practical terms, the natural logarithm is essential for understanding phenomena that grow or decrease at a rate proportional to their size—like radioactive decay, interest rates, or population growth.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable we're solving for stands alone on one side of the equation. This is a fundamental step in algebra that we employ to find the value of unknowns. When it comes to isolating a variable in an equation, we perform inverse operations such as subtraction or addition, and division or multiplication to 'move' other numbers or variables away from the variable of interest.

Handle With Care

When dealing with exponential equations, this process often involves additional skills like understanding the properties of exponents and logarithms. To isolate a variable like \(h\) in the exponential equation from our example, we first aim to get the exponential expression by itself on one side of the equation. After that, in the case of natural logarithms, we apply the ln to both sides of the equation because it allows us to 'bring down' the exponent, thereby isolating \(h\). Knowing how to skillfully perform this operation enables students to solve a wide range of problems in mathematics, physics, engineering, and even finance, making it an invaluable part of a problem solver's toolkit.

Density of Seawater

Density is defined as mass per unit volume and in the context of seawater, it is usually expressed in lb/ft³ (pounds per cubic foot). The density of seawater is not constant; it varies depending on several factors like salinity, temperature, and depth. Salinity and temperature are typically the primary influences on surface seawater density. However, at greater depths, pressure becomes a more significant factor and the density of seawater increases with depth due to compression.

Real-World Application

Understanding the density of seawater is crucial in several areas including oceanography, naval engineering, and climate science. For instance, it directly affects water buoyancy, which is vital for the design of submarines and ships. Additionally, density variations contribute to ocean currents that play a central role in distributing heat around the Earth's climate system. The equation from our original exercise shows that the density of seawater increases exponentially with depth, a fundamental concept for students who will study marine biology, geophysics, or environmental science. By grasping the connection between the depth and density of seawater, and how to mathematically determine it, students gain not only theoretical knowledge but also a practical understanding that applies to real-world observations and technologies.