Question

Danika and Sofie are staring up at a helicopter. Danika is directly east of the helicopter and Sofie is directlv west. The anale of elevation from Danika to the

helicopter is 31°. The angle of elevation from Sofie to the helicopter is 10°
Danika and Sofie are 100 feet apart. Find the height of the helicopter above the
ground. Round to two decimal places as necessary

Answer 1

Height of helicopter above ground = 13.63 feet

Explanation:

See attached image to support explanation

B is the point vertically below where the helicopter is located

s is the distance from Sofie to B

d is the distance from Danika to B

We have
s + d = 100 ==> s = 100 -d

h is the height of the helicopter from the base

Both triangles are right triangles

For a right triangle,
tan x = Side opposite hypotenuse/Side adjacent to hypotenuse
where x is the angle between the side adjacent to the hypotenuse

Using this information for both triangles we get

`\tan 10 = (h)/(s)\\or\\h = s ( \tan 10)\\\\\tan 31 = (h)/(d)\\or\\h = d (\tan31)\\\\`

Therefore
s (tan 10) = d ( tan 31)

But s = 100 -d:
(100-d) tan 10 = d ( tan 31)

100(tan 10) - d(tan 10) = d (tan 31)

Switch sides:
d (tan 31) = 100 (tan 10) - d( tan 10)

Add d (tan 10) to both sides:
d (tan 31) + d ·(tan 10) = 100(tan 10)

d(tan 31 + tan 10) = 100 (tan 10)

`d = (100(\tan 10))/(\tan 31 + \tan 10)`

Using a calculator we can compute right side as

`(100(\tan 10))/(\tan 31 + \tan 10) = 22.6878`

So
d = 22.6878

Plug this value of d into h = d sin 31 to get
h = d (tan 31)

h = 22.688 (0.6009)

h = 13.6322....

= 13.63 rounded to 2 decimal places

Height of helicopter above ground = 13.63 feet