ÐÏࡱá>þÿ 35þÿÿÿ2ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á� ø¿Ñbjbjoíoí $R ‡ ‡Ñÿÿÿÿÿÿ·°°sssss‡‡‡‡‡$«¼‡q&g}}}}XXX—¢9zsXXXX\Œss}}Û+ŒŒŒ`s}s}ŒXŒŒssŒ�¾)O~ʇvŒ(ìA0qŒ³Œ³ŒŒŒ€s XXXÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°‰ 9:    Case 2:Explore the area if the goat’s stake was placed by the silo’s door. First, we can easily calculate the area of the semi-circle:  Diagram 5: The Semicircle  EMBED Equation.3  Now, similar to the case with the barn, we must be careful as we calculate the remaining area. As the tether begins to move around the circumference of the silo, the goat’s grazing radius continually “shrinks”. The grazing radius shrinks based upon the distance of the silo’s circumference passed. While using integral calculus might be an easy way to determine the remaining area around the silo, this problem is an excellent opportunity to develop the fundamental ideas behind Calculus. Let us use Excel to begin to try to develop some of the elements behind Calculus. Since the grazing area will be symmetrical, we will only calculate half of the grazing area beyond the semicircle. Therefore, our journey can begin with calculating a “quarter” of the grazing area and then subtracting the corresponding half of the silo (where the goat can’t graze). This is seen in Diagram 6. While we could write out the equations necessary to do this crude approximation, it will be helpful to use an Excel file, see below: Sum of Sectors4620.41 Silo's Area157.08Grazing Area Approximation4463.33 Grazing Area *28926.66Semicircle9240.82Total Grazing Area18167.48  Diagram 6: First approximation of the Silo’s Grazing Area Next, realizing that we may get a better approximation, we evaluate two sections of the grazing area beyond the silo. See Diagram 7:  Diagram 7: Second approximation of the Silo’s Grazing Area Since we have more than one sector, we begin to notice the usefulness of Excel. We can develop various columns to organize our information. Our first column simply indicates what sector we are describing mathematically. The second column indicates the angle of the sector, and notice that the angle’s we are using provide an overestimation for the area (this is seen when the angle for the first row is equal to 0). The third column calculates the amount of the tether that is wrapped around the silo. The fourth column then determines at a given point, with the knowledge of the length of the tether wrapped around the silo, what is the radius of the tether from the silo. Now, before going any further, our motivation to find the area is to find the area of the individual sectors, sum them, and then subtract off half of the silo in order to calculate grazing area. But, it is important to realize that the radius of the tether from the silo is not necessarily equivalent to radius used to calculate the area of the sector. The radius used to approximate the area of the sector is the sum of the radius of the tether from the silo and the corresponding chord from the silo. However, in order to calculate the length of the chord, we needed to calculate the “height” of the chord (i.e. the distance from the center of the silo to the midpoint of the chord). Therefore, the fifth column calculated the height of the chord allowing the sixth column to calculate the length of the chord. Using the length of the chord and the radius of the tether from the silo, in column 7 we can find their sum to be the radius of the given sector. Quickly, column 8 calculates the area of the sector. The final portions of the excel spreadsheet sum all of the sectors, calculate the area of the silo, and then find the difference of the sum of the sectors from the area of the silo to produce a portion of the grazing area. Recall, this area must be multiplied by 2 and added to our original calculation for the semicircle. Because of the amount of columns, the excel spreadsheet has been linked for further viewing. The approximate grazing area using this method is shown in some of the final calculations from Excel: Sum of Sectors4160.70Silo's Area157.08Approximated Grazing Area4003.62 Grazing Area *28007.23 Semicircle Area9240.82 Total Grazing Area17248.05We can see using an additional sector to approximate this area, we move closer to a more precise estimate of the area the goat can graze. Let us see if we can refine it even more. What if we look at four sectors within this quarter of a “circle”. See Diagram 8 below:  Diagram 8: Third Approximation of the Goat’s Grazing Area About the Silo The summary from the Excel calculations indicates a more refined approximation for the goat’s grazing area about the silo: Sum of Sectors3930.01Silo's Area157.08Total Area For Grazing3772.93Grazing Area *27545.85 Semicircle Area9240.82 Total Grazing Area16786.68Consider what would happen if we continued our process – wouldn’t this be an interesting problem to introduce a delta? 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