Question

A boat heading out to sea starts out at

Point A, at a horizontal distance of 734
feet from a lighthouse/the shore. From
that point, the boat's crew measures the
angle of elevation to the lighthouse's
beacon-light from that point to be 14°.
At some later time, the crew measures
the angle of elevation from point B to
be 6°. Find the distance from point A to
point B. Round your answer to the
nearest foot if necessary.

Answer 1

Answer: 1007 feet

Step-by-step explanation

Refer to the diagram below. Point C is the base of the lighthouse. Point D is the top of the lighthouse.

x = distance from A to B

h = height of the lighthouse = distance from C to D

Focus on right triangle ACD.

tan(angle) = opposite/adjacent

tan(A) = CD/AC

tan(14) = h/734

h = 734*tan(14)

Next, focus your attention on right triangle BCD

tan(angle) = opposite/adjacent

tan(B) = CD/BC

tan(B) = CD/(BA + AC)

tan(6) = h/(x + 734)

tan(6) = 734*tan(14)/(x + 734)

(x + 734)*tan(6) = 734*tan(14)

x*tan(6) + 734*tan(6) = 734*tan(14)

x*tan(6) = 734*tan(14) - 734*tan(6)

x*tan(6) = 734*(tan(14) - tan(6))

x = 734*(tan(14) - tan(6))/tan(6)

x = 1007.192955967 approximately

x = 1007 when rounding to the nearest foot