Question

the pilot of an airplane flying at an altitude of 3000 feet sights 2 ships traveling in the same direction as the plane. the angle of depression of the further ship is 20 degrees and the angle of depression to the other ship is 35 degrees. find the distance between the 2 ships

Answer 1

The distance between the two ships can be determined using the tangent of the angles of depression and the altitude at which the plane is flying, resulting in two right-angled triangles with a common vertical side. By solving for the horizontal distances to each ship and subtracting them, we find the distance between the ships.

Step-by-step explanation:

The question involves using trigonometry to find the distance between two ships seen from an airplane using the angles of depression. When a pilot at an altitude of 3000 feet spots two ships with angles of depression at 20 degrees and 35 degrees, we can use right-angled triangles to calculate the horizontal distances from the plane to the ships. These triangles share a common vertical side (the altitude of the plane which is 3000 feet), but each has a different angle at the pilot's position, leading to two different horizontal distances.

First, we calculate the horizontal distance to the nearer ship:

1. Let d1 be the distance to the nearer ship.
   
2. Using the tangent of the angle of depression,
   tangent(35 degrees) = altitude / d1,
   we can solve for d1.
3. Now, let's compute the distance to the further ship, calling this distance d2.
   tangent(20 degrees) = altitude / d2.
4. Solving both equations for d1 and d2 will give us the horizontal distances to each ship from the pilot's perspective.
5. The difference between the two distances, d2 - d1, is the distance between the two ships.

Final calculations will involve a calculator to get the tangent values and to solve the equations for d1 and d2.

Answer 2

Alright, for this question it is important to remember the Law of Sines, which is sin*a/A = sin*b/B = sin*c/C, in which the lowercase letters are the angles and the uppercase letters are the sides opposing the angles.

Before we use that, however, we need to set up the picture, and as we do that, we can also find all of the angles we will need. Based on this question, we know that the farthest boat (boat 1) is 20 degrees below the plane's horizon and that the other boat (boat 2) is 35 degrees below the plane's horizon. From this, we can make triangles to connect the plane to the boats to give us some more info about the question. (Picture 1)

Now we have set up the triangles, we can see that there are two 90 degree angles that will help us find out the degrees of the remaining angles. We can do this because we know that the sum of the angles in a triangle always equals 180 degrees. (Picture 2)

Now, we can begin to use the Law of Sines. We will start with the triangle that connects the plane to the closest boat (boat 2) because we know that the plane is 3000 ft above sea level. from that, we can begin to set up our first proportion (picture 3). Next, we solve for x which means cross multiplying to get 3000*sin(90) = x*sin(55) which leaves us with x being roughly 3662.323

Finally, we put the number we just found into the drawing to have a little breather step before continuing (picture 4). For the last part, we are focusing on the middle triangle, which will give us the distance between the two boats when we are done. We use the Law of Sines once again to set up the proportions (picture 5) before cross multiplying again to get 3662.323*sin(35)=y*sin(20), which should simplify to get y to equal roughly 6141.808 ft, which should be the distance between the two boats!