A space shuttle pilot flying over the suez canal finds that the angle of depression to one end of the canal is 38.25 degrees and the angle of depression to the other end is 52.75 degrees. If the canal is 100.6 miles long, find the altitude of the space shuttle
To find the altitude of the space shuttle, we can use trigonometry.
Let's consider the triangle formed by the space shuttle, one end of the canal, and the center of the Earth. We can call the altitude of the space shuttle "h" (in miles).
From the given information, we know that the angle of depression to one end of the canal is 38.25 degrees. This means that the angle between the line of sight from the shuttle to the end of the canal and the horizontal plane is 38.25 degrees. Similarly, the angle of depression to the other end of the canal is 52.75 degrees. This means the angle between the line of sight from the shuttle to the other end of the canal and the horizontal plane is 52.75 degrees.
Since the angles are measured with respect to the horizontal plane, we can use the tangent function to determine the altitude of the shuttle.
First, let's find the distance from the shuttle to one end of the canal. We'll call this distance "x" (in miles). To find "x," we can use the tangent of the angle 38.25 degrees:
tan(38.25) = h / x
Rearranging the equation, we have:
x = h / tan(38.25)
Similarly, we can find the distance from the shuttle to the other end of the canal, let's call it "y" (in miles):
y = h / tan(52.75)
Since the canal is 100.6 miles long, we know that the sum of "x" and "y" will be equal to 100.6 miles:
x + y = 100.6
Substituting the expressions for "x" and "y" that we found earlier, we have:
h / tan(38.25) + h / tan(52.75) = 100.6
Now we can solve this equation to find the value of "h," which represents the altitude of the space shuttle.