Question

A surveyor sees a circular planetarium through an angle that measures 60°.33 ftA circle has an unlabeled center point. From a point outside the circle, on the left, two segments extend to touch the circle, and they form an angle.The upper segment goes up and right to the top left of the circle.The lower segment goes down and right to the bottom left of the circle, at a location labeled "Door". This segment is labeled 33 ft.The angle formed between the segments is 60°.If the surveyor is 33 ft from the door, what is the diameter (in ft) of the planetarium?ft

Answer 1

Solution:

Given the figure below:

To evaluate the diameter of the planetarium, we have

where OD is the radius.

step 1: In the triangle OAD, identify the sides.

Thus,

$\begin{gathered} OA\Rightarrow hypotenuse \\ AD\Rightarrow adjacent \\ OD\Rightarrow opposite \end{gathered}$

step 2: Evaluate OD, using trigonometric ratios.

From trigonometric ratios:

$\tan\theta\text{=}\frac{opposite}{adjacent}$

In this case, θ is 30, adjacent is 33.

Thus,

$\begin{gathered} \tan30=\frac{OD}{AD} \\ \Rightarrow\frac{\sqrt{3}}{3}=\frac{OD}{33} \\ cross-multiply, \\ OD=\frac{33\sqrt{3}}{3} \\ \Rightarrow OD=11\sqrt{3\text{ }} \end{gathered}$

step 3: Evaluate the diameter of the planetarium.

The diameter is evaluated as

$\begin{gathered} diameter=2\times radius \\ =2\times OD \\ =2\times11\sqrt{3} \\ \Rightarrow diameter=22\sqrt{3\text{ }}\text{ } \end{gathered}$

Thus, the diameter of the planetarium, in ft, is

$22\sqrt{3}$