Question

1. A signal tower has two lights that are 30 feet apart from each other, one directly above the other. From

a boat, the angle of elevation of the lower light is 14 degrees, and the angle of elevation yo the upper

light is measured to be 32 degrees.

a) Find to the nearest foot the distance from the boat to the lower light.

b) The boat and the foot of the tower are in the same horizontal plane. Find to the nearest foot the

distance from the boat to the foot of the tower.

Answer 1

It is given that the distance between the two lights is 30 feet.

The angle of elevation of the upper light from the boat, ∠ ABP = 32 degree

The angle of elevation of the lower light from the boat, ∠ ABQ = 14 degree

∴ ∠ PBQ = ∠ ABP - ∠ ABQ

= 32 - 14

= 18 degree

In triangle PAB

∠ PAB = 180 - (90 + 32)

= 58 degree

Similarly in triangle AQB

∠ AQB = 180 - (90 + 14)

= 76 degree

Now considering triangle PBQ and applying the sine theorem,

`$(PQ)/(\sin 18)=(BQ)/(\sin 58)$`

we know PQ = 30 feet

∴
`$BQ =(\sin 58)/(\sin 18) * 30$`

= 82.33 feet

Again taking the right triangle ABQ and applying sine theorem,

`$(BQ)/(\sin 90)=(AB)/(\sin 76)$`

`$(82.33)/(\sin 90)=(AB)/(\sin 76)$`

∴
`$AB = 82.33 * \sin 76$`

= 79.88 ft

= 80 feet