Question

A boat is heading towards a lighthouse, where Luis is watching from a vertical distance of 107 feet above the water. Luis measures an angle of depression to the boat at point A to be 10 degrees. At some later time,Luis takes another measurement and finds the angle or depression to the boat(now at point B) to be 39 degrees. Find distance from point A to point B. Round your answer to the nearest foot of necessary.

Answer 1

Using trigonometry, the distance between points A and B is determined by calculating the tangent of the angles of depression at both points. The calculated distance is approximately 474 feet, matching the correct answer.

The problem involves using trigonometry, specifically the tangent function, to determine the distance between points A and B. Let d be the distance from point A to the boat at point B.

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case:

1. For angle of depression 10 degrees at point A:

`\[ \tan(10^\circ) = (107)/(d) \]`

2. For angle of depression 39 degrees at point B:

`\[ \tan(39^\circ) = (107)/(d) \]`

Now, solve for d in both equations.

1. For point A:

`\[ d = (107)/(\tan(10^\circ)) \]`

2. For point B:

`\[ d = (107)/(\tan(39^\circ)) \]`

Calculating these values:

`\[ d_A \approx (107)/(\tan(10^\circ)) \approx (107)/(0.176327) \approx 606.54 \]`

`\[ d_B \approx (107)/(\tan(39^\circ)) \approx (107)/(0.809784) \approx 132.11 \]`

The distance from point A to point B is the difference between these two distances:

`\[ \text{Distance from A to B} \approx d_A - d_B \approx 606.54 - 132.11 \approx 474.43 \]`

Rounding to the nearest foot, the calculated answer is approximately 474 feet.