I am puzzled by this and I don't see where i am making a mistake. must be over looking something or looking at too much. nevertheless give it shot and see who can come up with the right answer..

A radar unit indicates that the distance between its antenna and a plane is 37,417 feet.
It also indicates that the plane is 30,000 feet north and 10,000 feet east of the antenna.
What is the altitude of the plane?

 

Neglecting the curvature of the earth, the distance along the ground of a point underneath the plane is:

SQRT(30,000^2 + 10,000^) = 31,623 ft. from the antenna

and the altitude is:

ALT = SQRT(37,417^2 - 31,623^2) = 20,000 ft AGL

 

Take a look at the article on slant range - http://www.radartutorial.eu/18.explanations/ex30.en.html