The Top of the Billboard

Question

The top of a billboard that is mounted on a base is 60 feet above the ground. At a point 25 feet from the foot of the base, the measure of the angle of elevation to the top of the base is one-half the measure of the angle of elevation to the top of the billboard.

A. Let theta be the measure of the angle of elevation to the top of the billboard . Find the cos theta and tan 1/2 theta.

B. Find the height of the base and the height of the billboard.

Answer 1

Sometimes the hardest part is trying to figure out what's going on. Draw a picture and things fall in place:

Looking only at the big triangle (top of billboard):

You can calculate θ by saying tan(θ) = 60/25 so θ = tan^-1(60/25) = 67.38°

Since the smaller angle (to the top of the base) is 1/2θ, then it is 67.38°/2 = 33.69°

Now, using the smaller triangle (top of base):

You can now say tan(33.69) = x/25 or x = 16 2/3 ft (≈ 16.67 ft)

The billboard is 60 - 16 2/3 = 43 1/3 ft (≈ 43.33 ft)

Answer 2

given: c=(a^2+b^2)^.5

side "b"=60 =65

side "a"=25 b/sinβ=65

side "c"=? 60/ " "

gamma "γ"=90° β=67.38°

"β"=? cosβ=.3846........

side"b"=top of billboard (given)

beta2 "β2"=?

"b2"=25

"α2" alpha2=beta/2=67.38/2=33.69° (given)

"a2"=?

"γ2"=90°

a2/sinα2=c2=b2/β2

sin90°=1

90-33.69=56.31°=β2

25/sin56.31=30.046=a2/.555

a2=16.66...............base height