H a n d o u t 2 p 3 L e s s o n 5 MATHEMATICS Problems on Conic Sections 13 You

H a n d o u t 2 p 3 l e s s o n 5 mathematics

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H a n d o u t 2 p . 3 L e s s o n 5 (MATHEMATICS) Problems on Conic Sections 13. You can think of a rectangle as a square that’s been stretched out in one direction, without doing anything in the perpendicular direction. This type of stretching is called a dilation. Using a ruler, measure the dimensions of the square and the rectangle in the figure above. How much was the square stretched horizontally to make the rectangle? Does it make more sense to describe this stretching amount as something you add to the square’s base, or as something you multiply the base by? By the way, the stretching amount is called the “scale factor.” 14. Can you consider a squeeze (instead of a stretch) to be a dilation? What sort of number would the scale factor be for a squeeze? 15. What effect does doubling the base of a triangle have on its area? What about doubling its height? 16. In the graphic below, what is the ratio of the area of the unshaded triangle to the area of the shaded triangle?
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19. Notice that you were not asked to actually use a basketball to model the sun in the previous activity. How big would your scale model of the solar system have to be if you did use a basketball to model the sun? 20. How would your model of the planetary orbits change if you included Pluto? 21. Two dilations.What happens to the area of a triangle if you double the base and double the height? What if you double the base but reduce the height to half its original value? 22. What effect does doubling the radius have on the area of a circle? 23. You can visualize an ellipse as a circle that has been dilated in one direction. Because of this, an ellipse has two radiuses instead of one: a large radius and a small radius. H a n d o u t 2 p . 4 L e s s o n 5 (MATHEMATICS) Problems on Conic Sections In the diagram here, the large radius is and small radius is . (These are also sometimes called the “semi-major axis” and “semi-minor axis,” respectively. But those terms are a mouthful, so using large and small radius is easier.) Using a ruler, find the values of and , in centimeters. If the ellipse was formed from a circle by dilating it horizontally, what was the scale factor? What if the dilation had been vertical instead? 24. Based on the fact that the area of a circle is , you might expect the area of an ellipse to be something like . This, in fact, actually is the correct formula. Can you use a dilation to explain why this formula works?
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