Math for Liberal Arts video. MAT 110: Chapter 10 Notes Geometry - - PowerPoint PPT Presentation

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Math for Liberal Arts MAT 110: Chapter 10 Notes

Geometry & Fractals David J. Gisch

Overview of Chapter 10

• We look at basic area and volume formulas.
• We will discuss the ideas of dimensionality.
• We will watch a video about Fractals.

▫ You will need to take notes and write a short paper about the video.

Points, Lines, and Planes

• A geometric point is imagined

to have zero size.

• A geometric line is formed by

connecting two points along the shortest possible path.

• Line segm ents are pieces of a

line.

• A geometric plane is a perfectly

flat surface that has infinite length and width, but no thickness.

Dimension

The dim ension of an object can be thought of as the number

• f independent directions in which you could move if you

were on the object. We can also think about dimension by the number of coordinates required to locate a point.

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Don’ t Limit Dimensions!!

• In Geometry we stick to 3D or less. It makes sense

because that is all that we can visually see.

• Dimensions do not have to be physical locations.
• For example, I collect data on my students.

▫ Age (16-90) ▫ GPA (0-4.0) ▫ Grade in my Class (0-100) ▫ Height (inches) ▫ Male/Female (0=Male, 1=Female)

• Each of these inputs can be thought of as a dimension.

Multiple Dimensions Dimensions and Measurement

Exam ples Dim ension Units Length Area Surface Area Volume

Role of Dimension

Example 10.1: A box is increased so it is 3 times larger. How do the following change? Length Area Volume

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Role of Dimension

Example 10.2: A pool has a floor with an area of 150 square

• feet. It’s volume is 1200 cubic feet (or 8,977 gallons). What

happens if the owner wants a pool that is twice as big? Area of Floor Volume

Parts of a Circle

Circles

• All points on a circle are located

at the same distance—the radius—from the circle’s center.

• The diam eter of a circle is twice

its radius.

Plane Geometry

A polygon is any closed shape in the plane made from straight line segments. A regular polygon is a polygon in which all the sides have the same length and all interior angles are equal.

Perimeter and Area

Use 3.1415 or use the button your calculator.

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Three-Dimensional Geometry Area and Perimeter

Example 10.3: Find the area and perimeter of the following window.

Volume

Example 10.3: Which can of soup holds more?

Area and Perimeter

Example 10.4: A local high school wants to install a new border around the field of a track. They also wish to fertilize the grass to make it look good for an upcoming meet. Give them the need information (in feet) to buy brick bordering and fertilizer.

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Pythagorean Theorem

The Pythagorean theorem applies only to right triangles (those with one 90 angle). For a right triangle with side lengths a, b, and c, in which c is the longest side (or hypotenuse), the Pythagorean theorem states that a2 + b2 = c2

a b c

Pythagorean Theorem

Example 10.5: A piece of drywall needs to be cut for the side of the stairs shown below. What is the area of that piece? Also, if a piece of trim needs to be put along the hypotenuse, how long should it be?

Pitch, Grade, S lope

Type of Value Calculation Exam ple Pitch Ratio : 9: 12 3: 4 Grade Percent

• %

9 12 .75 75% Slope Fraction

• 9

12 3 4

We use the stairs as examples in the table above.

Pitch, Grade, S lope

Example 10.6: The board of tourism in a state brags that its bike trails are for the whole family with a maximum of 3%. What is the corresponding pitch and slope? Put this information into practical terms.

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A little background for Tuesday’s Movie

Fractals

• Fractals are multiple (LOTS!!!!) of repetitions of a

pattern.

• Often the repetition creates smaller and smaller pieces

for a more textured/real looking image. Begin with a line segment L0 of length 1. Then generate L1 with the following steps: 1. Divide the line segment L0 into three equal pieces.

• 2. Remove the middle piece.
• 3. Replace the middle piece with

two segments of the same length arranged as two sides of an equilateral triangle. Repeat the steps for each line segment of the current figure to generate the next figure.

The S nowflake Curve The S nowflake Island

The snowflake island is a region (island) bounded by three snowflake curves.

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S ierpinski Triangle

The Sierpinski triangle is produced by starting with a solid black equilateral triangle and iterating with the following rule: For each black triangle in the current figure, connect the m idpoints of the sides and rem ove the resulting inner triangle.

S ierpinski Triangle The Mandelbrot S et

The first fractal. Created by Benoit Mandelbrot.

Two Views of Barnsley’s Fern

Fractals are a huge part of computer generated imaging.

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Fractal Images