Problem 45 - Single Variable Calculus

Question

Avoiding a Collision Upon spotting a disabled and stationary boat, the driver of a speedboat took evasive action. Suppose that the disabled boat is located at the point \((0,2)\) in an \(x y\) -coordinate system (both scales measured in miles) and the path of the speedboat is described by the graph of \(f(x)=(x+1) / \sqrt{x}\) a. Find an expression \(D(x)\) that gives the distance between the speedboat and the disabled boat. b. Plot the graph of \(D\), and use it to determine how close the speedboat came to the disabled boat before it changed its path.

Step-by-Step Solution

Answer

The distance between the speedboat and the disabled boat is given by the expression \(D(x) = \sqrt{x^2 + (\frac{x + 1}{\sqrt{x}} - 2)^2}\). By plotting the graph of D(x) and looking for the minimum distance, it was determined that the speedboat came closest to the disabled boat at a distance of 1 mile before changing its path.

Step 1: Define the positions of the boats

The disabled boat is located at point (0,2), so its coordinates are: Disabled boat: (0, 2) The speedboat is moving along the path described by the function f(x) = (x + 1) / sqrt(x), so its coordinates are: Speedboat: (x, f(x)) Now we can use the distance formula to find the distance between the two boats.

Step 2: Use the distance formula

The distance formula is given by: \(D(x) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) In this case, the coordinates for the two boats are (0, 2) and (x, f(x)). Therefore: \(D(x) = \sqrt{(x - 0)^2 + (f(x) - 2)^2}\)

Step 3: Substitute the function for the speedboat's path

We know that the path of the speedboat is given by \(f(x) = \frac{x + 1}{\sqrt{x}}\). We can substitute this into the distance formula: \(D(x) = \sqrt{(x - 0)^2 + (\frac{x + 1}{\sqrt{x}} - 2)^2}\) Simplify the expression inside the square root: \(D(x) = \sqrt{x^2 + (\frac{x + 1}{\sqrt{x}} - 2)^2}\)

Step 4: Plot the graph of D(x)

To plot the graph of D(x), we can use a graphing calculator or software like Desmos. You will see that D(x) approaches its minimum value as x approaches 1.

Step 5: Find the minimum distance

Looking at the graph, we see that the minimum distance occurs at approximately x = 1. We can find the corresponding value of D(x) by plugging in x = 1 into the distance formula: \(D(1) = \sqrt{1^2 + (\frac{1 + 1}{\sqrt{1}} - 2)^2}\) \(D(1) = \sqrt{1 + (2 - 2)^2}\) \(D(1) = \sqrt{1}\) D(1) = 1 The closest the speedboat came to the disabled boat was 1 mile before changing its path.

Key Concepts

Coordinate System

The coordinate system is an essential concept in mathematics, especially in calculus and analytical geometry. It's the foundation upon which we plot graphs, locate points, and discuss geometric figures. Typically, we use a two-dimensional Cartesian coordinate system, composed of a horizontal axis (often referred to as the x-axis) and a vertical axis (the y-axis). Points are located using pairs of numbers, known as coordinates, which represent their distances from two perpendicular lines called axes. These axes intersect at a point called the origin, which has the coordinates (0, 0).

In our exercise, the disabled boat is located at point (0,2), which means it is two units above the origin along the y-axis, while its x-coordinate is at the origin. Understanding the coordinate system is vital for visualizing and solving problems that involve the positions of objects in space.

Plotting Graphs

Plotting graphs is a powerful tool when it comes to visualizing functions and the relationship between variables. Through the representation of functions on a graph, we can analyze their behavior, identify features such as intercepts, maxima, minima, and asymptotes, and solve problems that pertain to real-world scenarios.

In the context of the exercise, once the function \(f(x) = \frac{x + 1}{\sqrt{x}}\) that describes the speedboat's path is determined, graphing it over a range of x-values will depict the movement of the speedboat over time or space. The graph helps us to see the distance function \(D(x)\) and where it reaches its minimum, allowing for the calculation of how close the speedboat came to the stationary boat before changing its path. Graphs are an intuitive way to understand complex relationships and are widely used in various fields, including physics, engineering, economics, and statistics.

Calculating Minimum Distance

Calculating the minimum distance between two points is a problem that often arises in calculus and optimization. It involves finding the point at which the distance between two curves or a point and a curve is the smallest. The problem typically requires the use of the distance formula, which in two dimensions is \(D(x) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

To apply this to our scenario, we take the distance formula and analyze it as a function of one variable, which represents the position of the moving object (in this case, the speedboat). By finding the derivative of this function and setting it equal to zero, we can locate the minimum distance. Alternatively, graphical methods can be used to approximate where this minimum occurs, as seen in Step 4 of the textbook solution. Knowing how to calculate minimum distance has real-world applications in fields like navigation, physics, and computer graphics.

Function Substitution

Function substitution is a crucial technique within calculus used to simplify complex expressions and solve equations. It involves replacing a part of an equation with another function that is equal to that part, which can make a problem easier to manage and solve. Substitution can also refer to replacing a variable with a specific value to find the outcome of a function at that point.

In our speedboat exercise, the substitution involves inserting \(f(x) = \frac{x + 1}{\sqrt{x}}\) into the distance formula's y-component for the position of the speedboat. Through this substitution, we create a new function \(D(x)\) that exclusively depends on the x-coordinate of the speedboat's position. This newly formed function \(D(x)\) can then be analyzed to find out the minimum distance between the speedboat and the stationary boat. Substitution not only simplifies problems but often opens the door to analytical solutions that might otherwise be inaccessible.