Question 709202: From a plane 2000m above sea level, the pilot observes two ships in line due west. The angle of depression of the two ships are 60 degrees and 70 degrees. How far apart are the ships?
Can you also please help me with illustrating this? I have a hard time illustrating triangles. Thanks.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! For the diagram...- Draw two parallel horizontal lines. The top one will represent the path of the plane. The bottom one represents the surface of the water.
- Pick a point on the top line (closer to the end of the line than the middle). This will represent the plane. Let's label it "P".
- Pick two separate points on the bottom line (near the other end of the line from where P is). Make sure there is space between the two points. These points will represent the ships. Label the ship closer to P as "X" and the ship farther away as "Y".
- Draw four line segments:
- From P to X
- From P to Y
- From X straight up to the upper line. Label the upper endpoint of this segment as "Q".
- From Y straight up to the upper line. Label the upper endpoint of this segment as "R".
Since the two lines are horizontal and segments XQ and YR are vertical, XQ and YR are perpendicular to the two horizontal lines. So draw the "box" indicator of a right angle at each point where XQ and YR intersect the horizontal lines. (IOW at X, Y, Q and R).- Since the lines are horizontal lines are parallel and since distance between lines is measured perpendicularly and since XQ and YR are perpendicular, both XQ and YR are 2000m in length. Write 2000 next to XQ and YR to indicate their length.
- The angles of depression (which are always angles from a horizontal) are angles RPY and QPX. If you've been following these directions, angle RPY is the smaller angle. So label it 60 degrees. Angle QPX is the 70 degree angle. But since angle RPY is inside angle QPX it will not be easy to label angle QPX. Perhaps it would be better just to make a note outside the diagram that the measure of angle QPX is 70 degrees.
Our diagram is now complete. It should have two right triangles, RPY and QPX, and a rectangle, XYRQ. There is another triangle, PXY. But it is not a right triangle and we only know one part of this triangle. (Angle XPY is the difference between angle RPY and QPX so it is 10 degrees.) We need three parts, including at least one side, of a non-right triangle to find all the other parts. So if possible we will be using just the two right triangles and the rectangle to solve the problem.
The problem asks us to find the distance between the ships. In our diagram this should be the length of XY. XY is not a side of either of the right triangles so we will not be able to solve for it directly. We will have to find other lengths first and then use them to find XY.
XY is a side of the rectangle. So its length is the same as the opposite side, QR. If we can find QR then we have also found XY.
QR is the difference between PR and PQ. Since PR and PQ are sides of our right triangles and since we know the angles of depression and the vertical sides of these triangles we can use Trig to find PR and PQ.
In triangle RPY the vertical side, YR, is opposite to the the angle of depression, angle RPY. And the side we want to find is the adjacent side to the angle of depression. The Trig functions that involve a ratio of opposite and adjacent are tan and cot. Since we do not have a button on our calculator for cot, we will use tan:
 (Note: PR is not P times R. It is a single number/variable that is the length of side PR.)
Solving for PR. Multiplying both sides by PR we get:
Dividing both sides by tan(60):
60 is a special angle so we could find the tan exactly without a calculator. (It's  .) But we will use a calculator and get a decimal anyway since- PR is not the answer to our problem; and
- we will be subtracting PQ from PR; and
- we do not have a special angle in triangle QPX so we will have to use our calculator to find PQ; and
- there's not much point in using an exact value (with a square root) when you have to subtract a decimal approximation from it to find your answer.
Replacing tan(60) with its decimal approximation:
 (Note: Feel free to use more decimal places.)
Dividing:
PR = 1154.7
We will repeat this for PQ. The only difference is that the angle is 70, not 60:
PQ = 727.9
QR, as noted earlier, is the difference between PR and PQ:
QR = PR - PQ
QR = 1154.7 - 727.9
QR = 426.8
And since QR and XY are opposite sides in a rectangle, XY must be 426.8, too. So the distance between the ships is approximately 426.8 meters. (Note: You might get a number slightly different from this is you used more (or fewer) decimal places for your tan's.)
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