Two groves

Two groves A, 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:

$$\begin{gathered} b=5004\text{ m } \\ a=2600\text{ m } \\ B=53+45\mathrm{/}60={\displaystyle \frac{215}{4}}=53.75^{\circ } \\ {c}^2=a^2+c^2\text{−}2ac\cos \beta \\ k=\cos B^{\circ }=\cos 53.75^{\circ }=0.59131 \\ {b}^2=a^2+c^2-2\ast a\ast c\ast k \\ 5004^2=2600^2+c^2-2\cdot 2600\cdot c\cdot 0.591309648364 \\ -c^2+3074.81c+18280016=0 \\ {c}^2-3074.81c-18280016=0 \\ p=1;q=-3074.81;r=-18280016 \\ D=q^2-4pr=3074.81^2-4\cdot 1\cdot (-18280016)=82574521.5907 \\ D>0 \\ {c}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{3074.81\pm \sqrt{82574521.59}}{2}} \\ {c}_{1,2}=1537.40508575\pm 4543.52620744 \\ {c}_1=6080.93129318 \\ {c}_2=-3006.12112169 \\ \text{ Factored form of the equation: } \\ (c-6080.93129318)(c+3006.12112169)=0 \\ c=c_1=6080.9313\text{ m} \end{gathered}$$

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