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A goat is tethered by a 6m rope to the outside corner of a shed measuring 4m by 5m in a grassy field. What area of grass can the goat graze?

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Introduction

TETHERED GOAT

 A goat is tethered by a 6m rope to the outside corner of a shed measuring 4m by 5m in a grassy field.

 What area of grass can the goat graze?

PROBLEM STATEMENTimage00.png

As stated above the question is asking to find the maximum amount of area of grass the goat can graze. To tackle this problem we will be dividing the area of grass into sections that we can deal with. We will add

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Middle

image40.pngimage39.pngimage38.png

image02.pngimage02.pngimage02.png

                                           4m

image41.pngimage42.png

image44.png

                                                         ttty

                                                                                                         Length of rope is 10m    

  1. Find the equations for the arcs of the circles which bound region (as shaded in diagram 3 below).

Diagram(3)

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Conclusion

2 = 28.27m2

Area of blue and green quarter circle = ¼ *pi*52 = 19.635m2

Since we have used the area of green twice. We need to find what is the actual area of this green part.

We know that the green area is approximately a quarter circle of radius 1m. So if we can work the approximate value of the green area, we can then find the total area of the grass the goat can eat.

Total area of lawn eaten by the goat is

=(3/4*pi*102) + (1/4*pi*62) + (1/4*pi*52) – (1/4*pi* 12)

                                      =  235.619449 + 28.27433388 + 19.63495409 – 0.785                                                            

                                      = 283.528737 – 0.785

                                    T0TAL AREA = 282.74m2

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