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 in the distance to be 17° and 11°, respectively. How big is the island (i.e. how long is the island)?

Answer 1

To find the length of the island, we use the angles of depression and the height of the cliff to set up right triangles and then apply trigonometric functions to solve for the distances to the near and far sides of the island. The difference between these distances gives us the size of the island.

Step-by-step explanation:

To calculate the length of the island, we can use the concept of angles of depression in conjunction with basic trigonometric principles. By drawing two right triangles from the observer down to the near and far sides of the island, we can use the tangent function, which is the ratio of the opposite side (length of the island from the observer to each end) over the adjacent side (the height of the cliff).

Let's denote the distances from the observer to the near and far sides of the island as d_near and d_far respectively. Using the angles of depression (17° and 11°) along with the height of the cliff (320 feet), we have:

• tan(17°) = opp/adj —> tan(17°) = d_near/320
• tan(11°) = opp/adj —> tan(11°) = d_far/320

By solving these equations for d_near and d_far, we can determine the length of the island:

• d_near = tan(17°) × 320
• d_far = tan(11°) × 320

The size of the island is then length of the island = d_far - d_near.