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Math Help - Area of Goat Grazing

  1. #1
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    Area of Goat Grazing

    A goat is tied to the corner of 7 foot square barn. It has a 20 foot long rope. What is the area of its grazing? I thought it would be 300*pi+169*pi/4-1/4*36*pi. However I got about 1179 and the correct answer rounds to 1176. Did I make a mistake or did I do it wrong. I said 300*pi because 3/4 sector of a 20 foot radius circle, 169*pi/4 because of two quarter sectors of 13 radius circles, and -1/4*36*pi, because the two radius thirteen circles overlap each other by 6 foot radii in a 90 degree circular arc.
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  2. #2
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    Quote Originally Posted by empireruler View Post
    A goat is tied to the corner of 7 foot square barn. It has a 20 foot long rope. What is the area of its grazing? I thought it would be 300*pi+169*pi/4-1/4*36*pi. However I got about 1179 and the correct answer rounds to 1176. Did I make a mistake or did I do it wrong. I said 300*pi because 3/4 sector of a 20 foot radius circle, 169*pi/4 because of two quarter sectors of 13 radius circles, and -1/4*36*pi, because the two radius thirteen circles overlap each other by 6 foot radii in a 90 degree circular arc.
    1. I've attached a drawing.

    2. There are 3 different areas to calculate:

    Threequarter of a circle with radius r = 20: a_1=\frac34 \cdot \pi \cdot 20^2

    Two sectors with radius r = 13 and the central angle \alpha = \arctan \left(\frac{12}5\right) \approx 67.38^\circ
    Thus the area is a_2 = 2 \cdot \frac{67.38}{360} \cdot \pi \cdot 13^2

    Two triangles with the base length of 7 and the height of 5: a_3 = 2 \cdot \frac12 \cdot 7 \cdot 5

    The complete area is a = a_1+a_2+a_3

    I've got a \approx 1176.222
    Attached Thumbnails Attached Thumbnails Area of Goat Grazing-ziegegrasflaeche.png  
    Last edited by earboth; June 19th 2010 at 06:40 AM. Reason: fixing LaTeX
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