Question

A statue is mounted on top of a 21 foot hill. From the base of the hill to where you are standing is 57feet and the statue subtends an angle of 7.1° to where you are standing. Find the height of the statue.

Answer 1

We will have situation as shown in figure in attachment.

Here OH is the hill of height 21 feet.

Statute is HS, so lets its height be h, so HS = h

Distance OA = 57 feet, observer distance from base of hill

Then ∠HAS = 7.1° (angle subtended by statue HS at point A)

First we wil find ∠OAH in right ΔOAH

so let ∠OAH be Ф

We will use $tanФ =\frac{opposite-side}{adjacent -side} = \frac{OH}{OA}$

$tanФ =\frac{21}{57}$

so $Ф = tan^-1( \frac{21}{57} )$

Ф = 20.22°

Now in right ΔOAS, we have

∠OAS = Ф + 7.1° = 20.22° +7.1° = 27.32°

so tan (27.32°) = $\frac{OS}{OA} =\frac{21+x}{57}$

$0.516581 =\frac{21+x}{57}$

Now we will solve for x here

So multiply both sides by 57

$0.516581 \times 57 =\frac{21+x}{57}\times 57$

$29.445117 =21 + x$

29.445117 -21 =21 + x -21

8.445 = x

So height of statue is 8.45 feet.