Goat

The fenced flower bed has the shape of a regular hexagon. The tops are formed by fence posts. The fence around the flowerbed measures 60 m. A goat is tied to one of the pillars from the outside and grazes on the surrounding meadow (the goat should not enter the flowerbed). The twine measures 24 m. How many square meters of the meadow can a goat graze?

Correct answer:

S =  1428.3775 m²

Step-by-step explanation:

$$\begin{gathered} o=60\text{m} \\ r=24\text{m} \\ a=o/6=60/6=10\text{m} \\ r>a \\ \alpha =360-2\cdot 60=240{}^{\circ } \\ {S}_1=\pi \cdot r^2\cdot \frac{\alpha }{360}=3.1416\cdot 24^2\cdot \frac{240}{360}\doteq 1206.3716{\text{m}}^2 \\ \beta =60{}^{\circ } \\ {S}_2=\pi \cdot (r-a{)}^2\cdot \frac{\beta }{360}=3.1416\cdot (24-10{)}^2\cdot \frac{60}{360}\doteq 102.6254{\text{m}}^2 \\ {S}_3=\pi \cdot (r-2\cdot a{)}^2\cdot \frac{\beta }{360}=3.1416\cdot (24-2\cdot 10{)}^2\cdot \frac{60}{360}\doteq 8.3776{\text{m}}^2 \\ S=S_1+2\cdot S_2+2\cdot S_3=1206.3716+2\cdot 102.6254+2\cdot 8.3776\doteq 1428.3775{\text{m}}^2 \\ \text{ Verifying Solution: } \\ M=\pi \cdot r^2=3.1416\cdot 24^2\doteq 1809.5574{\text{m}}^2 \\ M>S \end{gathered}$$

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