Two groves

Two groves A and B are separated by a forest. Both are visible from the hunting grove C, which is connected to both by direct roads. What will be the length of the projected road from A to B if AC = 5004 m, BC = 2600 m, and angle ABC = 53° 45'?

Correct answer:

c =  6080.9313 m

Step-by-step explanation:

$$\begin{gathered} b=5004\text{m} \\ a=2600\text{m} \\ B=53+45/60=\frac{215}{4}=53.75{}^{\circ } \\ {c}^2=a^2+c^2-2ac\cos \beta \\ k=\cos B=\cos 53.75°=0.59131 \\ {b}^2=a^2+c^2-2\cdot a\cdot c\cdot k \\ 5004^2=2600^2+c^2-2\cdot 2600\cdot c\cdot 0.59130964836358 \\ -c^2+3074.81c+18280016=0 \\ {c}^2-3074.81c-18280016=0 \\ p=1;q=-3074.81;r=-18280016 \\ D=q^2-4pr=3074.81^2-4\cdot 1\cdot (-18280016)=82574521.590713 \\ D>0 \\ {c}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{3074.81\pm \sqrt{82574521.59}}{2} \\ {c}_{1,2}=1537.405086\pm 4543.526207 \\ {c}_1=6080.931293184 \\ {c}_2=-3006.121121692 \\ c=c_1=6080.9313=6080.9313\text{m} \end{gathered}$$

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