The farmer

The farmer sees the back fence of the land, which is 50 m long at a viewing angle of 30 degrees. It is 92 m away from one end of the fence. How far is it from the other end of the fence?

Final Answer:

c₁ =  99.2703 m
c₂ =  60.0784 m

Step-by-step explanation:

$$\begin{gathered} a=50\text{m} \\ \alpha =30{}^{\circ } \\ b=92\text{m} \\ {a}^2=b^2+c^2-2\cdot b\cdot c\cdot \cos \alpha \\ k=2\cdot b\cdot \cos \alpha =2\cdot b\cdot \cos 30°=2\cdot 92\cdot \cos 30°=2\cdot 92\cdot 0.866025=159.34867\text{m} \\ {a}^2=b^2+c^2-k\cdot c \\ 50^2=92^2+c^2-159.34867429634\cdot c \\ -c^2+159.349c-5964=0 \\ {c}^2-159.349c+5964=0 \\ p=1;q=-159.349;r=5964 \\ D=q^2-4pr=159.349^2-4\cdot 1\cdot 5964=1536.0000000009 \\ D>0 \\ {c}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{159.35\pm \sqrt{1536}}{2} \\ {c}_{1,2}=79.674337\pm 19.595918 \\ {c}_1=99.27025509 \\ {c}_1=c_2=60.078419206=99.2703\text{m} \end{gathered}$$

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