A boat is heading towards a lighthouse, whose beacon-light is 102 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 7 , before they draw closer. They measure the angle of elevation a second time from point b at some later time to be 12. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

351 ft (nearest foot)

Step-by-step explanation:

Model the problem as a right triangle, where the horizontal leg is the water (where the boat is at the vertex with the hypotenuse), and the vertical leg is the lighthouse.  The angle of elevation is between the boat and the top of the lighthouse. (see attached diagram)

Horizontal distance from point A to the lighthouse

Using the trig ratio tan(x) = O/A where x is the angle, O is the side opposite to the angle and A is the side adjacent to the angle:

Given:

⇒ tan(7) = 102/A

⇒ A = 102/tan(7)

⇒ A = 830.7233357... ft

Horizontal distance from point B to the lighthouse

Given:

⇒ tan(12) = 102/B

⇒ B = 102/tan(12)

⇒ B = 479.8722712... ft

Therefore, distance from A to B:

⇒ distance = A - B

⇒ distance = 830.7233357... - 479.8722712...

⇒ distance = 350.8510645...

⇒ distance = 351 ft (nearest foot)

area = length x width

length = 2 x width

32 = 2 x width x width

32 = 2 x width^2

36 = width ^2  

width = 6

length = 12

area is 32

perimeter = 2 x 12 + 2 x 6 = 24 + 12 = 36

step-by-step explanation: