Find the height of the antenna to the nearest foot.

Question

A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is 65 degrees. He then walks 60 feet further away and observes that the angle of elevation to the top of the antenna is 40 degrees.

Find the height of the antenna to the nearest foot.

Answer 1

If h is the height and xis the distance from the base of the antenna to the first position then

h / x = tan 65°

h / (x+60) = tan 40°

So x = h / tan 65°

So h / [ h / tan 65° + 60] = tan 40°

h = h tan 40 / tan 65 + 60 tan 40

h ( 1 - tan 40 / tan 65) = 60 tan 40

h = 60 tan 40 / [ 1 - (tan 40 / tan 65)\

Answer 2

Draw and label a diagram

h represents the height of the tower

x represents the distance from the tower to first sightings.

A is the location of the first sighting

B is the location of the second sighting.

tan 65° = h / x

x tan 65° = h

x = h / tan 65°

tan 40° = h / (60 + x)

(60 + x) tan 40° = h

60 tan 40° + x tan 40° = h

x tan 40° = h - 60 tan 40°

x = (h - 60 tan 40°) / tan 40°

Set the two right sides equal and solve for h.