Question

From the top of a lighthouse 51 feet above sea level, the angle of depression to

a small boat is 11.9°. How far from the foot of the lighthouse is the boat?
The boat is about
feet away.
(Round to the nearest foot as needed.)

Answer 1

Using the tangent function, the distance from the lighthouse to the boat is approximately 232 feet. Rounding to the nearest foot, the boat is about 232 feet away.

To find the distance from the foot of the lighthouse to the boat, trigonometry, specifically the tangent function, can be used. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Let d be the distance from the foot of the lighthouse to the boat. In this case, the angle of depression is 11.9 degrees, and the height of the lighthouse is 51 feet. So, the tangent of the angle is given by:

`\[ \tan(11.9^\circ) = (51)/(d) \]`

To solve for d, we rearrange the equation:

`\[ d = (51)/(\tan(11.9^\circ)) \]`

Now, we can calculate the distance:

`\[ d \approx (51)/(\tan(11.9^\circ)) \approx (51)/(0.219512) \approx 232.38 \]`

Therefore, the boat is about 232 feet away from the foot of the lighthouse. Rounding to the nearest foot, the boat is approximately 232 feet away.