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Chapter 6: Problem 107

Write a short paragraph explaining why parametric equations are useful.

Short Answer

Expert verified
Parametric equations are useful as they provide a flexible and succinct way to describe complex geometric shapes and their positions. They're vital in fields like physics, for studying motion, engineering for system modeling, and computer graphics for object transformations.

Step by step solution

01

Defining Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points that make up a geometric object such as a curve or surface, as functions of a variable, called a parameter. In simpler terms, they allow us to describe geometric objects in a more flexible way using a common parameter.
02

Explaining the usefulness of Parametric Equations

Parametric equations are beneficial because they allow us to express complex geometric relationships in a simple way. They provide a more flexible method to describe geometric shapes and their positions, which conventional Cartesian equations may not handle effectively or succinctly. Furthermore, they are essential in several fields like physics, where motion and change are constantly studied; engineering, for equipment and system modeling; computer graphics for object transformations, and many more applications.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$x^{2}+y^{2}-2 a y=0$$

A quarterback releases a pass at a height of 7 feet above the playing field, and a receiver catches the football at a height of 4 feet,30 yards directly downfield. The pass is released at an angle of \(35^{\circ}\) with the horizontal. (a) Write a set of parametric equations for the path of the football. (See Exercises 93 and 94 .) (b) Find the speed of the football when it is released. (c) Use a graphing utility to graph the path of the football and approximate its maximum height. (d) Find the time the receiver has to position himself after the quarterback releases the football.

The equation \(r=\frac{e p}{1 \pm e \sin \theta}\) is the equation of an ellipse with \(e<1 .\) What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

Think About It \(\quad\) Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. A. \(y=a(x-h)^{2}+k, \quad a \neq 0\) B. \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) C. \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

Find the distance between the point and the line. Point \((2,3)\) Line \(3 x+y=1\)
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