Question

Use a graphing utility to graph $y=10\ln \left(\frac{10+\sqrt{100-x^2}}{10}\right)-\sqrt{100-x^2}$ over the interval $(0,10]$. This graph is called a tractrix or pursuit curve. Use your school's library, the Internet, or some other reference source to find information about a tractrix. Explain how such a curve can arise in a real-life setting.

Short answer

The curve is a track called a tractrix or pursuit curve. This curve occurs, for instance, in mechanical engineering where it’s involved in the design of gears and belts in mechanical drives due to its ability to ensure a smooth and effective transmission of power.

Step-by-step solution

Graph the Function

Use a graphing utility to plot the equation $y=10\ln \left(\frac{10+\sqrt{100-x^2}}{10}\right)-\sqrt{100-x^2}$. For this, you need to substitute a range of x-values within the interval (0, 10] into the equation and then plot the corresponding y-values.

Understanding the Tractrix Curve

The curve produced from the graph of the function is known as a tractrix or pursuit curve. It gets its name from the Latin verb 'trahere', meaning 'to pull', because it represents the path of an object being pulled in a straight line. Observe the curve from the graph plotted in step 1. You will notice that this curve is a path generated by a point (called a tractrix) that moves such that its distance from a given straight line (called the locus) remains constant.

Real-life Application Scenario

For the real-life application of a tractrix curve, search the school's library, the Internet or use some other reference sources. A common scenario where this curve is used is in mechanical engineering, specifically in the design of gears and belts in motor vehicles. In such a scenario, the gear teeth follow a tractrix profile to ensure smooth and efficient transmission of power.