Question

An observer standing on a cliff 320 feet above the ocean measured angles of depression of the near and far sides of an island to be 16° and 10° respectively. How long is the island to the nearest foot?

Answer 1

The length of the island is 699 feet.

Explanation:

Let the distance from the foot of the cliff to the near side be represented by x, and the distance from the foot of the cliff to the far side be represented by y.

Applying the trigonometric function to calculate the value of x;

Tan θ =
`(opposite)/(adjacent)`

Tan
`16^(o)` =
`(320)/(x)`

0.2867 =
`(320)/(x)`

x =
`(320)/(0.2867)`

= 1116.1493

x = 1116.15 feet

The distance from the foot of the cliff to the near side of the island is 1116.15 feet.

To determine the value of y;

Tan θ =
`(opposite)/(adjacent)`

Tan
`10^(o)` =
`(320)/(y)`

0.1763 =
`(320)/(y)`

y =
`(320)/(0.1763)`

= 1815.0879

y = 1815.09 feet

The distance from the foot of the cliff to the far side of the island is 1815.09 feet.

The length of the island = y - x

= 1815.09 - 1116.15

= 698.94

The length of the island is 699 feet.