math problem angle of elevation

Question

Two lines are tied between the railing of a tall ship and the main mast. They are tied to the same point on the railing. The angles of elevation of the lines are 68◦ and 52◦. If the total length of the two lines is 80 feet, how far is the railing from the main mast? (Give the final answer to the nearest foot.)

Answer 1

Draw two right triangles emanating from the same point on the rail to different points on the mast. Let the 52 degree elevation triangle have a hypotenuse of x feet and the 68 degree elevation triangle have a hypotenuse of (80 - x ) feet. Let y be the common base which is also the distance from railing to the main mast that we are looking for.

cos 52 = y/x and cos 68 = y/(80 - x)

y = cos 52(x)

y also = cos 68(80 - x)

So cos 52 x = cos 68(80 - x)

.616x = .375(80) -.375x

.991x = 30

x = 30.27

y = cos 52(30.27)

y = 18.636 feet rounded to 19 feet.