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Two observers are
500ft apart on opposite sides of a flagpole. The angles of elevation from the observers to the top of the pole are
18^(@) and
20^(@). Find the height of the flagpole.
The flagpole is
◻ ft righ.
(Round to the nearest tenth as needed.)

Two observers are $500 \mathrm{ft}$ apart on opposite sides of a flagpole. The angles of elevation from the observers to the top of the pole are $18^{\circ}$ and $20^{\circ}$ . Find the height of the flagpole. $\newline$ The flagpole is $\square$ ft righ. $\newline$ (Round to the nearest tenth as needed.)

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Circles: word problems

Full solution

Q. Two observers are

500 \mathrm{ft}

apart on opposite sides of a flagpole. The angles of elevation from the observers to the top of the pole are

18^{\circ}

and

20^{\circ}

. Find the height of the flagpole.

\newline

The flagpole is

\square

ft righ.

\newline

(Round to the nearest tenth as needed.)

Set up using trigonometry: Set up the problem using trigonometry. $\newline$ We have two right triangles formed by the flagpole and the lines of sight from the observers to the top of the flagpole. We can use the tangent function, which is the ratio of the opposite side (height of the flagpole) to the adjacent side (distance from the observer to the base of the flagpole).
Write tangent equations: Write the equations using the tangent of the angles. $\newline$ Let $h$ be the height of the flagpole. For the first observer: $\newline$ $\tan(18°) = \frac{h}{500}$ $\newline$ For the second observer: $\newline$ $\tan(20°) = \frac{h}{500}$
Solve for height: Solve for $h$ in both equations. $\newline$ From the first observer: $\newline$ $h = 500 \times \tan(18°)$ $\newline$ From the second observer: $\newline$ $h = 500 \times \tan(20°)$
Calculate using tangent values: Calculate the height using the tangent values. $\newline$ Using a calculator, we find: $\newline$ $h = 500 \times \tan(18°) \approx 500 \times 0.3249 \approx 162.45$ $\newline$ $h = 500 \times \tan(20°) \approx 500 \times 0.3639 \approx 181.95$
Identify setup mistake: Realize there is a mistake in the setup. $\newline$ We should not have the same distance for both observers since they are on opposite sides of the flagpole. The total distance between the observers is $500\,\text{ft}$ , so we need to split this distance based on the angles. This is a mistake in the setup of the problem.

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