A village has a circular wall around it, and the wall has four...

A village has a circular wall around it, and the wall has four gates pointing north, south east and west. A tree stands outside the village, 16 m north of the north gate, and it can be just seen appearing on the horizon from a point 48 m east of the south gate. What is the diameter in meters, of the wall that surrounds the village? A village has a circular wall around it, and the wall has four gates pointing north, south east and west. A tree stands

Correct answer

Step-by-step solution

Let radius equals to

r

\begin{aligned} & A B=\sqrt{A O^2-O B^2}=\sqrt{(16+r)^2-r^2}=\sqrt{256+32 r} \\ & A D^2+C D^2=A C^2 \end{aligned}

\begin{aligned} & \Rightarrow(48)^2+(2 r+16)^2=(48+\sqrt{256+32 r})^2 \\ & \Rightarrow r^2+8 r=24 \sqrt{256+32 r} \\ & \Rightarrow r(r+8)_{-}=24 \times 4 \sqrt{2} \sqrt{r+8} \\ & \Rightarrow r \sqrt{r+8}=24 \sqrt{32} \\ & \Rightarrow r=24 \end{aligned}

Diameter equal to 48

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