Question

A surveyor has determined that a mountain is h = 2440 ft high. From the top of the mountain he measures the angles of depression to two landmarks at the base of the mountain and finds them to be 42° and c = 37°. (Observe that these are the same as the angles of elevation from the landmarks as shown in the figure below.) The angle between the lines of sight to the landmarks is 68°. Calculate the distance between the two landmarks. (Round your answer to the nearest integer.)

Answer 1

First, we are going to draw a diagram of the situation.

Next, from triangle ABC we can find the length of $a$ using the sine trig function:
$sin( \alpha )= \frac{opposite.side}{hypotenuse}$
$sin(42)= \frac{2440}{h}$
$h= \frac{2440}{sin(42)}$
$h=3646.52$

Now, since we know that the angle between the lines of sight to the landmarks is 68°, we can use the law of sines in triangle ABD to find the distance $a$ between the two landmarks:
$\frac{a}{sin(68)} = \frac{3646.52}{sin(37}$
$a= \frac{3646.52sin(68)}{sin(37)}$
$a=5617.996$

We can conclude that the distance between the two landmarks is 5618 feet.