The Geometry of a Circle - By: Dennis Kapatos

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$By: Melissa Hanel Math B/Grade 11 5 Day Unit Plan Technologies ...$

Problems with Watering Crops Name: ________________ Date: _________________ Answers in Boxes in Red Farmer Bob is using a new high-tech watering system which uses a computer and many high powered sprinklers to water his crops for him. To organize his land, Bob decides to divide his entire field into 100 meter square sections. Sprinklers are placed strategically in order to water as much of the land as possible without overlap (see figure). 1) What do you notice about this arrangement of sprinklers? Some areas don’t get watered. The property line for Bob’s Land is in Red. The 100 meter square sections are in blue. The red dots are the sprinkler heads with their spraying area shown in green. 2) Estimate: About what percent of Bob’s fields do you think are not getting any water? Responses Vary.
Notice that some areas of Bob’s field get no water at all. The computer uses the center sprinkler as the origin, that is, it assigns it the coordinates, (0,0). All other points on Bob’s field, such as the point (100,0), are measured from this point. (0,0) (100,0) Notice that that this point, (90,60), isn’t getting any water. 3) A water sensor placed in the ground tells Bob that the coordinates of one of the points which is not receiving water are (90,60). Fortunately, the sprinklers can be programmed to spray different distances. What distance should Bob tell the sprinkler at point (0,0) to spray in order for it to reach this point? Obviously you could set the sprinkler to spray a much larger area then needed, but his would waste water. What is the exact distance from the sprinkler to the dry point? (Show all work.) d = √( (90-0) 2 + (60-0) 2 ) ≈ 108 meters 4) There are other points in this area as well. What distance should Bob tell the sprinkler to spray in order to water any given point (x,y)? √( (x) 2 + (y) 2 ) Distance = _____________ 5) Bob’s computer refers to distance as the letter r and it doesn’t like square root signs either. How could you rewrite the equation above so Bob can enter the distance into the computer? __________=___________________ r 2 = (x) 2 + (y) 2 r = 5, cntr = (0,0) The above equation will give the sprinkler a new spraying radius that will reach any point (x,y) on the coordinate grid. It is the general equation of a circle with its center located at the point (0,0) and with a radius of r. 6) What is the radius and center of the circle formed by the equation 5 2 = x 2 + y 2 ? 7) What are they for the circle 36 = x 2 + y 2 ? r = 6, cntr = (0,0) 8) What is the equation of a circle with center (0,0) and radius 10? With radius 4? 100 = x 2 + y 2 16 = x 2 + y 2
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Page 19 and 20: 8. Class will discus why they think
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The Geometry of a Circle - By: Dennis Kapatos

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