Question

A person moves from the origin along the positive \(y\) -axis pulling a weight at the end of a 12 -meter rope. Initially, the weight is located at the point (12,0) . (a) In Exercise 64 of Section 6.6 , it was shown that the path of the weight is modeled by the rectangular equation $$ y=-12 \ln \left(\frac{12-\sqrt{144-x^{2}}}{x}\right)-\sqrt{144-x^{2}} $$ where \(0

Short answer

Step 1 and Step 2 involve plotting an equation and parametric equations using a graphing utility. Step 3 asks for a comparison between these plots. Step 4 demonstrates the tangent line's y-intercept and point of tangency remains constant, regardless of the location of the tangency point.

Step-by-step solution

Graph the Rectangular Equation

Using a graphing utility, graph the equation \(y=-12 \ln \left(\frac{12-\sqrt{144-x^{2}}}{x}\right)-\sqrt{144-x^{2}}\) for \(0<x \leq 12\).

Graph the Parametric Equations

Now, plot the parametric equations \(x=12 \operatorname{sech} \frac{t}{12}\) and \(y=t-12 \tanh \frac{t}{12}\), where \(t \geq 0\), on the same graphing interface.

Compare the Graphs

Upon analyzing the two graphs, discuss the differences and similarities. Highlight the key features, shapes, and patterns from the two solutions - visually compare the tracks of both approaches. Look closely and analyze which graph provides a better representation of the weight's path.

Verify the Constant Distance

Lastly, to confirm that the distance between the y-intercept of the tangent line and the point of tangency remains the same regardless of the location of the tangency point, we will need to use calculus. The derivative of the parametric equations gives the tangent line's slope. Using this, the equation of the tangent line can be found. The y-intercept of this line and the point of tangency can be determined. Once these are established, the distance between these two points needs to be calculated. Even without explicit values for the tangency point's location, you should be able to show that this value is constant and independent of the location of the tangent point.

Key concepts

Graphing Utility

Utilizing a graphing utility provides a visual representation of complex mathematical concepts. It allows students to plot and analyze the behavior of different functions, which is especially helpful in understanding the intricacies of mathematical equations. When dealing with the provided rectangular equation, a graphing utility can be employed to illustrate the trajectory. The exercise suggests graphing the equation \[ y=-12 \ln \left(\frac{12-\sqrt{144-x^{2}}}{x}\right)-\sqrt{144-x^{2}} \] for \(0
To get the best out of the graphing utility, it’s crucial to adjust the viewing window to include all relevant areas of the graph, paying special attention to the range \(0

Tractrix Path

The tractrix path shares a fascinating attribute: it is the trajectory taken by an object that is being dragged in a straight line from a starting point, while maintaining a fixed distance from another point. The given exercise introduces a scenario where a weight is pulled along the y-axis from the origin, while it is tethered to a 12-meter rope initially located at point (12,0). In mathematical terms, this motion can be modeled by both rectangular and parametric equations.

The benefit of using a parametric representation, which includes equations \(x=12 \operatorname{sech} \frac{t}{12}\) and \[y=t-12 \tanh \frac{t}{12}\], where \(t \geq 0\), is that it often provides a clearer picture of the object's motion since time ‘t’ is accounted for explicitly. The tractrix path is particularly notable for its applications in real-life scenarios such as in the design of gear teeth or in the analysis of certain types of motion in physics. Furthermore, when graphed, it shows the constant length of the restraint (the rope, in this scenario) and the peculiar nature of the curve produced.

Calculus Applications

Calculus is an essential tool in analyzing and understanding motion, rates of change, and many other aspects of the physical world. In the context of this exercise, calculus applications are critical in verifying certain properties of the tractrix path. Part (c) of the exercise specifically requires the application of derivatives, which are fundamental in calculus, to establish that the distance from the y-intercept of the tangent line to the point of tangency remains constant irrespective of the location of the point of tangency.

By using the parametric equations to find the derivatives, it's possible to determine the slope of the tangent lines at different points along the trajectory. This involves taking the derivatives \(x'(t)\) and \(y'(t)\) and then analyzing the behavior of the tangent line using these slopes. The uniformity of the distance from the y-intercept to the tangent point can then be demonstrated through careful calculation, showcasing the power of calculus to reveal and prove properties of geometrical figures like the tractrix.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point without crossing it, and their slopes represent the instantaneous rate of change at that point. The examination of tangent lines related to the tractrix path reveals the dynamic relationship between the linear and curved aspects of the motion described in the exercise.

The tangent lines to a curve at various points can be determined by taking the derivative of the curve's equation. In this scenario, by finding the derivative of the parametric equations for the tractrix, one can compute the slope of the tangent line at any given point 't.' This slope is then used to form the equation of the tangent line. To realize the implication of part (c), the fact that the length of the rope remains constant during the movement along the tractrix path can also be correlated to the constant distance between the y-intercept of these tangent lines and their corresponding points of tangency. Understanding tangent lines and their properties are critical for many applications in physics, engineering, and even computer graphics, where the concept is used to simulate realistic movements and forces.

Tractrix Path

The tractrix path shares a fascinating attribute: it is the trajectory taken by an object that is being dragged in a straight line from a starting point, while maintaining a fixed distance from another point. The given exercise introduces a scenario where a weight is pulled along the y-axis from the origin, while it is tethered to a 12-meter rope initially located at point (12,0). In mathematical terms, this motion can be modeled by both rectangular and parametric equations.

The benefit of using a parametric representation, which includes equations \(x=12 \operatorname{sech} \frac{t}{12}\) and \[y=t-12 \tanh \frac{t}{12}\], where \(t \geq 0\), is that it often provides a clearer picture of the object's motion since time ‘t’ is accounted for explicitly. The tractrix path is particularly notable for its applications in real-life scenarios such as in the design of gear teeth or in the analysis of certain types of motion in physics. Furthermore, when graphed, it shows the constant length of the restraint (the rope, in this scenario) and the peculiar nature of the curve produced.

Calculus Applications

Calculus is an essential tool in analyzing and understanding motion, rates of change, and many other aspects of the physical world. In the context of this exercise, calculus applications are critical in verifying certain properties of the tractrix path. Part (c) of the exercise specifically requires the application of derivatives, which are fundamental in calculus, to establish that the distance from the y-intercept of the tangent line to the point of tangency remains constant irrespective of the location of the point of tangency.

By using the parametric equations to find the derivatives, it's possible to determine the slope of the tangent lines at different points along the trajectory. This involves taking the derivatives \(x'(t)\) and \(y'(t)\) and then analyzing the behavior of the tangent line using these slopes. The uniformity of the distance from the y-intercept to the tangent point can then be demonstrated through careful calculation, showcasing the power of calculus to reveal and prove properties of geometrical figures like the tractrix.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point without crossing it, and their slopes represent the instantaneous rate of change at that point. The examination of tangent lines related to the tractrix path reveals the dynamic relationship between the linear and curved aspects of the motion described in the exercise.

The tangent lines to a curve at various points can be determined by taking the derivative of the curve's equation. In this scenario, by finding the derivative of the parametric equations for the tractrix, one can compute the slope of the tangent line at any given point 't.' This slope is then used to form the equation of the tangent line. To realize the implication of part (c), the fact that the length of the rope remains constant during the movement along the tractrix path can also be correlated to the constant distance between the y-intercept of these tangent lines and their corresponding points of tangency. Understanding tangent lines and their properties are critical for many applications in physics, engineering, and even computer graphics, where the concept is used to simulate realistic movements and forces.