Problem 210 - Calculus Volume 1

Question

An anchor drags behind a boat according to the function \(y=24 e^{-x / 2}-24,\) where \(y\) represents the depth beneath the boat and \(x\) is the horizontal distance of the anchor from the back of the boat. If the anchor is 23 ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.

Step-by-Step Solution

Answer

You need to pull approximately 7.856 feet of rope to reach the anchor.

Step 1: Set Up the Equation

We need to find the distance \(x\) when \(y = -23\). Substitute \(y = -23\) into the given function: \[ -23 = 24e^{-x/2} - 24. \]

Step 2: Isolate Exponential Term

Add 24 to both sides of the equation to isolate the exponential term: \[ 1 = 24e^{-x/2}. \]

Step 3: Divide Both Sides by 24

Divide both sides by 24 to solve for the exponential expression: \[ \frac{1}{24} = e^{-x/2}. \]

Step 4: Take the Natural Logarithm

Take the natural logarithm of both sides to solve for \(-\frac{x}{2}\): \[ \ln\left(\frac{1}{24}\right) = -\frac{x}{2}. \]

Step 5: Solve for x

Multiply both sides of the logarithmic equation by \(-2\) to solve for \(x\): \[ x = -2 \ln\left(\frac{1}{24}\right). \] Calculate the value to get \[ x \approx 7.856. \]

Key Concepts

Natural Logarithm

The natural logarithm, often written as \( \ln \), is a special kind of logarithm. It's based on the number \( e \), approximately 2.71828, which is a constant similar to \( \pi \). The natural log is used extensively in math, especially when dealing with exponential functions, which involve the \( e\) constant. When solving equations involving exponentials, the natural log is an important tool to "undo" or "reverse" the exponential effect. If you have \( e^x \), applying the natural log will help you find the power that \( e \) was raised to. This is because \( \ln(e^x) = x \).

• Useful for solving equations of the form \( e^{f(x)} = g \).
• Allows you to take the exponent "out of" the function, turning it into regular algebra.
• Critical in calculus for differentiating and integrating functions involving \( e^x \).

By applying the natural log in our problem, we determined the change in distance \( x \), which was essential for finding the required rope length.

Equation Solving

Solving equations is all about finding the value of a variable that makes the equation true. In our problem with the anchor function, we start by setting the equation according to the depth provided. This gives the equation \( -23 = 24e^{-x/2} - 24 \).We then follow these steps:1. **Isolate the exponential term**: Add 24 to both sides to focus on the exponential expression. This step is crucial to make the exponential part the center of attention.2. **Divide by 24**: Simplifies the left-hand side to \( \frac{1}{24} = e^{-x/2} \). This makes it easier for us to apply logarithmic operations.Each step simplifies the equation, helping us gradually solve for \( x \). The final log operation simplifies exponential equations by helping us "get x out of the exponent". This is the beauty of algebraic manipulation, turning a complicated expression into something computationally simple.

Exponential Functions

Exponential functions are those written in the form \( y = a e^{bx} + c \). They have the characteristic of growing quickly and have a constant rate of growth or decay when expressed with the constant \( e \). In our problem, the function \( y = 24 e^{-x/2}-24 \) represents the depth of an anchor beneath a boat based on the distance.Key aspects of exponential functions:

• They describe many natural phenomena like population growth, radioactive decay, and cooling.
• In our case, it models how the anchor's depth changes with horizontal distance from the boat.
• The exponential component \( e^{-x/2} \) implies a decay; as the distance \( x \) increases, the value it affects decreases exponentially.

Understanding exponential functions is foundational in calculus, as they often lead to rates of change problems. The anchor problem is an example of applying exponential decay to determine a physical situation, the amount of rope needed for a given depth, illustrating how math models real-world contexts.