Answer:
1372 feet
Step-by-step explanation:
The horizontal distance between the lighthouse and the boat, the height of the lighthouse and the distance boat-lighthouse keeper form a right triangle, of which:
- The distance between keeper and boat is the hypothenuse
- The horizontal distance between boat and base of the lighthouse is the side adjacent to the angle of ![\theta=5^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3D5%5E%7B%5Ccirc%7D)
- The height of the lighthouse is the side opposite to the angle of ![\theta=5^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3D5%5E%7B%5Ccirc%7D)
So we can write:
![tan \theta = \frac{opposite}{adjacent}](https://tex.z-dn.net/?f=tan%20%5Ctheta%20%3D%20%5Cfrac%7Bopposite%7D%7Badjacent%7D)
where in this situation,
opposite = height of the lighthouse = 120 ft
Therefore, the length of the adjacent side is
![adjacent = \frac{opposite}{tan \theta}=\frac{120}{tan 5^{\circ}}=1372 ft](https://tex.z-dn.net/?f=adjacent%20%3D%20%5Cfrac%7Bopposite%7D%7Btan%20%5Ctheta%7D%3D%5Cfrac%7B120%7D%7Btan%205%5E%7B%5Ccirc%7D%7D%3D1372%20ft)
So, the distance between the boat and the base of the lighthouse is 1372 ft.