Modeling with Geometry 10A Discussion Paragraph 1 web 88. The Geometry of Ancient Cultures 89. Surveying and GIS 90. Platonic Solids 1 world 91. Geometry in the News 92. Geometric Idealizations 93. Circles and Polygons 94. Three Dimensional Objects Copyright © 2011 Pearson Education, Inc.

Problem Solving with Geometry Unit 10B Problem Solving with Geometry Copyright © 2011 Pearson Education, Inc.

Measuring Angles To measure angles more precisely, each degree is subdivided into 60 minutes of arc; and each minute is subdivided into 60 seconds of arc. Copyright © 2011 Pearson Education, Inc.

Fractional Degrees CN (1a-b) Convert a. Convert 3.6° into degrees, minutes, and seconds of an arc. b. Convert 30°33’31” into decimal form. Copyright © 2011 Pearson Education, Inc.

Latitude and Longitude We can locate any place on the Earth’s surface by its latitude and longitude. Copyright © 2011 Pearson Education, Inc.

Latitude and Longitude CN (2a-d) Answer each of the following questions, and explain your answers clearly. a. Suppose you could drill from Miami straight through the center of Earth and continue in a straight line out the other side of Earth. At what latitude and longitude would you emerge? b. Perhaps you’ve heard that if you dug a straight hole from the US through the center of the Earth, you’d come out in China. Is this true? c. Suppose you travel 1° of latitude north or south. How far have you traveled? (hint: the circumference of the Earth is 25,000 miles) d. Suppose you travel 1° of longitude east or west. How far have you traveled?

Angular Size and Distance The farther away an object is located from you, the smaller it will appear in angular size. Copyright © 2011 Pearson Education, Inc.

Angular Size and Distance CN (3a-b) A quarter is about 1 inch in diameter. a. Approx. how big will it look in angular size if you hold it 1 yard (36 inches) from your eye? The angular diameter of the Moon as seen from Earth is approx. 0.5°, and the Moon is approx. 380,000 km from Earth. b. What is the real diameter of the moon? Copyright © 2011 Pearson Education, Inc.

How Steep? CN (4a-c) Suppose a road has 100% grade. a. What is its slope? Pitch? What angle does it make with the horizontal? b. Which is steeper: a road with an 8% grade or a road with a pitch of 1 in 9? c. Which is steeper: a roof with a pitch of 2 in 12, or a roof with a pitch of 3 in 15? Copyright © 2011 Pearson Education, Inc.

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 Copyright © 2011 Pearson Education, Inc.

Distance Measurements CN (5a-b) Consider a map showing several city streets in a rectangular grid (10.30 on p.572). The individual city blocks are 1/8 of a mile in the east-west directions and 1/16 of a mile in the north-south direction. a. How far is the library from the subway along the path shown? b. How far is the library from the subway “as the crow flies” that is along a straight diagonal path)? Copyright © 2011 Pearson Education, Inc.

Similar Triangles Two triangles are similar if they have the same shape, but not necessarily the same size, meaning that one is a scaled-up or scaled-down version of the other. Copyright © 2011 Pearson Education, Inc.

Similar Triangles For two similar triangles, corresponding pairs of angles in each triangle are equal: the ratios of the side lengths in the two triangles are all equal: Copyright © 2011 Pearson Education, Inc.

Lot Size CN (6) Find the area, in acres, of the mountain lot show in Figure 10.31. 250 feet, 1200 feet property line 6. Area = Copyright © 2011 Pearson Education, Inc.

Practice with Similar Triangles CN (7a-b) Figure 10.33 shows two similar triangles. Find the lengths of the sides labeled a and c’. a. a = b. c’ = Copyright © 2011 Pearson Education, Inc.

Solar Access CN (8) Some cities have policies that prevent property owners from constructing new houses and additions that cast shadows on neighboring houses. The intent of these policies is to allow everyone access to the Sun for the use of solar energy devices. Consider the following solar access policy: On the shortest day of the year, a house cannot cast a noontime shadow that reaches farther than the shadow that would be cast by a 12 foot fence on the property line. Suppose your house is set back 30 feet from the north property line and that, on the shortest day of the year, a 12 foot fence on that property line would cast a noontime shadow 20 feet in length. 8. If you build and addition to your house, how high can the north sided of the remodeled house be under this policy?

Optimization Problems Optimization problems seek a “best possible” solution. Example: You have 200 meters of fence to enclose a corral. If you want the corral to have the greatest possible area, should you make it a circle or square? Square corral: Circular corral: To optimize the area of the corral, it should be a circle. Explain that the side length for a square is 200/4=50 m. For a circle, explain that since C=2πr, the radius of the circle is the total length of the fence (the circumference) divided by 2π. Copyright © 2011 Pearson Education, Inc.

Optimizing Area CN (9) You have 132 meters of fence that you plan to use to enclose a corral on a ranch. What shape do you choose if you want the corral to have the greatest possible area? 9. What is the area of this “optimized” corral? Copyright © 2011 Pearson Education, Inc.

Optimal Container Design CN (10a-b) You are designing a wooden crate that must have a volume of 2 cubic meters. The cost of the wood is $12 per square meter. a. What dimensions give the least expensive design? b. hat that optimal design, how much will the material for each crate cost? Copyright © 2011 Pearson Education, Inc.

Quick Quiz CN (11) Please answer the ten multiple choice questions on p. 576. Copyright © 2011 Pearson Education, Inc.

Homework 10B Discussion Paragraph 10A CN 1-11 p.576:1-14 1 web 98. Great Circles 99. Sphere Packing Copyright © 2011 Pearson Education, Inc.