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Chapter 10: Problem 52

Try drawing an ellipse as accurately as possible on a blackboard. How would a piece of string and two friends help this process?

Short Answer

Expert verified
Use a string and two foci points to draw an ellipse by holding the string taut and moving around.

Step by step solution

01

Understanding the Problem

An ellipse is a geometric shape that can be drawn using two fixed points called foci. The main task here is to use a piece of string to utilize these foci and draw an accurate ellipse on a blackboard.
02

Setting Up the Tools

Gather a long piece of string, two extra hands (your friends), and some chalk. The string should be longer than the distance between the two foci by twice the desired distance from a point on the ellipse to each focus (the major axis length).
03

Positioning the Foci

Choose two points on the blackboard to act as the foci of the ellipse. These points should be at an appropriate distance from each other depending on the desired shape and size of the ellipse.
04

Attaching the String

Tie the ends of the string to make a loop. Place the loop over the two foci. Hold the string taut by having your friends lightly hold the string at these focal points.
05

Drawing the Ellipse

Take a piece of chalk and use it to pull the string taut at a point along the string loop, forming a triangle with the two foci. Keeping the string taut, move the chalk carefully around, maintaining tension, until you come back to the starting point. This will trace out an ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Shapes
Geometric shapes are everywhere around us and form the foundation of geometry. Among them, an ellipse is a fascinating two-dimensional shape.
It appears as an elongated circle, with a symmetry that is pleasing to the eye. You can see ellipses in everyday objects like running tracks, the orbits of planets, and even the shadows cast by tilted circles.
Understanding the basic properties of geometric shapes helps us analyze and construct more complex systems in nature or art. The ellipse, with its simple yet elegant form, is key to many applications in math and beyond.
Foci
The concept of foci is integral to understanding ellipses. In mathematics, the plural of 'focus' is 'foci,' and for an ellipse, there are always two.
These are fixed points on the interior of the ellipse. What makes them special is the property that for any point on the ellipse, the sum of the distances to the two foci is constant.
  • Understanding foci helps in drawing perfect ellipses.
  • Ensures your construction is accurate and follows mathematical properties.
  • The foci influence the overall shape: the closer they are, the more circular the ellipse.
Being aware of these elements lets you construct ellipses confidently, knowing that the resulting shape encapsulates this essential property.
Ellipse Construction
Constructing an ellipse can be a fun activity involving practical tools like strings and teamwork. Here's how you can do it:
1. Select two points to act as your foci on a surface such as a blackboard. 2. Tie the ends of a string to form a loop, which should be larger than the distance between the foci. 3. Have two friends hold the string looped around the foci, ensuring it remains taut.
Next, use a piece of chalk to draw the ellipse:
  • Place the chalk so it keeps the string taut, forming a triangle with the foci.
  • Move the chalk around the foci, maintaining the tension, to trace the elliptical path.
  • Complete the ellipse by returning to the starting point.
This simple method leads to a precise ellipse using basic geometric principles, understanding the role of foci and the tension adjustment help achieve an accurate figure.
Major Axis
The major axis is a crucial part of an ellipse. It is the longest diameter that passes through the center of the ellipse, intersecting the foci.
Understanding the major axis helps in:
  • Visualizing the longest line that can be drawn through the ellipse.
  • Determining the overall size of the ellipse because the total length is twice the major axis鈥檚 half-length, called the semi-major axis.
  • Deciding how far the string should extend beyond the foci when creating an ellipse.
By knowing the length of the major axis, you can plan and draw an accurate ellipse, ensuring your creation reflects geometric rules and aesthetics accurately.

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Most popular questions from this chapter

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$25 x^{2}-120 x y+144 y^{2}-156 x-65 y=0$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sec t, \quad y=\tan ^{2} t, \quad 0 \leq t<\pi / 2$$

Spiral Path of a Dog A dog is tied to a circular tree trunk of radius \(1 \mathrm{ft}\) by a long leash. He has managed to wrap the entire leash around the tree while playing in the yard, and finds himself at the point \((1,0)\) in the figure. Seeing a squirrel, he runs around the tree counterclockwise, keeping the leash taut while chasing the intruder. (a) Show that parametric equations for the dog's path (called an involute of a circle) are $$x=\cos \theta+\theta \sin \theta \quad y=\sin \theta-\theta \cos \theta$$ (b) Graph the path of the dog for \(0 \leq \theta \leq 4 \pi\). CAN'T COPY THE GRAPH

Use a graphing device to draw the curve represented by the parametric equations. $$x=2 \sin t, \quad y=\cos 4 t$$

A Family of Confocal Conics Conics that share a focus are called confocal. Consider the family of conics that have a focus at \((0,1)\) and a vertex at the origin (see the figure). (a) Find equations of two different ellipses that have these properties. (b) Find equations of two different hyperbolas that have these properties. (c) Explain why only one parabola satisfies these properties. Find its equation. (d) Sketch the conics you found in parts (a), (b), and (c) on the same coordinate axes (for the hyperbolas, sketch the top branches only) (e) How are the ellipses and hyperbolas related to the parabola?
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