Question

# In a circus, a guy wire A is attached to the top of a 30-ft pole. Wire B is used for performers to walk up to the tight wire, 10 ft above the ground.

Circles
In a circus, a guy wire A is attached to the top of a 30-ft pole. Wire B is used for performers to walk up to the tight wire, 10 ft above the ground. Find the angle ϕ between the wires if they are attached to the ground 40 ft from the pole.

2021-06-01

Let x be the angle of the smaller right triangle and y be the angle of the larger right triangle so that:
$$\displaystyleϕ={y}−{x}$$
For the smaller right triangle with hypotenuse as guy wire B, we can use the tangent ratio to find x:
$$\displaystyle{\tan{{x}}}=\frac{{10}}{{40}}$$
$$\displaystyle{x}={{\tan}^{{-{{1}}}}{\left(\frac{{10}}{{40}}\right)}}$$
$$\displaystyle{x}\sim{14.04}°$$
For the larger right triangle with hypotenuse as guy wire A, we can use the tangent ratio to find y:
$$\displaystyle{\tan{{y}}}=\frac{{{30}+{10}}}{{40}}$$
$$\displaystyle{\tan{{y}}}={1}$$
$$y=\tan^{-1}*1$$
$$y=45^{\circ}$$
Hence, $$\displaystyleϕ={45}°−{14.04}°$$
$$\displaystyleϕ≈{30.96}°$$