Question

Distance \(\quad\) Two insects are crawling along different lines in three- space. At time \(t\) (in minutes), the first insect is at the point \((x, y, z)\) on the line \(x=6+t, \quad y=8-t, \quad z=3+t\). Also, at time \(t,\) the second insect is at the point \((x, y, z)\) on the line \(x=1+t, \quad y=2+t, \quad z=2 t\). Assume distances are given in inches. (a) Find the distance between the two insects at time \(t=0\). (b) Use a graphing utility to graph the distance between the insects from \(t=0\) to \(t=10\) (c) Using the graph from part (b), what can you conclude about the distance between the insects? (d) How close do the insects get?

Short answer

The short answer should include the distance between the insects at \(t=0\), a description of the graph showing how the distance changes from \(t=0\) to \(t=10\), a conclusion about their movement based on the graph, and the minimum point on the graph that represents the closer distance between the insects.

Step-by-step solution

Calculate the distance

First of all, calculate the distance between the insects at time \(t=0\). For that, substitute \(t=0\) in the coordinates of both insects and then use the formula for distance in 3-D space: \(d = \sqrt{{(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2}}\) . After substituting the values of \(x, y, z\) coordinates for both insects at \(t=0\), calculate the distance.

Graph the distance

The distance between the two insects is a function of time, \(d(t)\). So, to achieve this step, a function \(d(t)\) should be defined using the previously mentioned formula for the 3D distance but leaving \(t\) as a variable. Then instruct the graphing utility to plot this function from \(t=0\) to \(t=10\). The plot will show how the distance between the insects changes over the given time period.

Interpret the graph

After the graph is plotted, observe the shape of the graph to derive conclusions. If the graph is constantly increasing, it means the insects are moving apart. If it is decreasing, they are moving closer. If the graph has a minimum point within the range, it means the insects get closer at a certain point before moving apart again.

Find the closest distance

From the plot, find the minimum point of the graph, if any. This point represents the shortest distance between the insects throughout the given time period. It might be at a specific input time or it might be the lowest point displayed on the vertical axis if the insects were initially closer.

Key concepts

Distance Formula in Calculus

Understanding the distance formula in calculus is vital when analyzing the separation between points in three-dimensional space. To calculate the distance between two points, \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\), the formula is \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \].

This formula stems from the Pythagorean theorem, extended into three dimensions. It's a reliable tool for measuring straight-line distances, regardless of direction. For instance, if you're measuring the proximity of two moving insects, as in our exercise, you apply the distances formula at a given time to find the distance between them. Consistently using this formula can help students readily solve problems related to distances in not just math, but fields like physics and engineering as well.

Utilizing a Graphing Utility

Graphing utilities are invaluable in visualizing mathematical concepts, especially when dealing with complex functions or data. By translating the distance function \(d(t)\) into visual form from \(t=0\) to \(t=10\), you can get a clearer picture of how distance changes over time.

With these utilities, you can plot points, lines, functions, and even implicit relationships to explore the interplay of variables like time and distance. By graphing the distance between two points as they move through space, we not only get a curve that shows the distance over time but can also identify key features such as maxima, minima, and inflection points, offering deeper insight into the motion.

3D Coordinates

The concept of 3D coordinates extends our standard two-dimensional plane into the third dimension, allowing for a more comprehensive representation of space.

In a 3D coordinate system, each point is represented by an ordered triple \( (x, y, z) \) which corresponds to its position along the x, y, and z-axes. This three-axis system is essential for mapping out locations and movements in physical space—a critical skill in fields like robotics, architecture, and computer graphics. When working through our example with two insects, we used their changing 3D coordinates to calculate their distances at varying times, showcasing the applicability of this concept.

Rate of Change

The 'rate of change' in mathematics usually refers to how a quantity changes with respect to a change in another quantity, often time. In physics, this is analogous to speed or velocity when discussing movement. By examining the rate of change of the distance function \(d(t)\), we can infer not just how the distance between two points changes but also how quickly this change occurs.

In our insects' case, if the rate of change is positive, the insects are moving apart; if negative, they're getting closer. Understanding the rate at which distance changes provides critical insights into the nature of the movement, potential collisions, or, as highlighted by the exercise, the moment when the insects are closest.