Express the Distance d between the plane and the top of the control tower as a function of x?

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1 Answer
May 22, 2018

#d=90400#ft #+x^2#.

Explanation:

What we have in this diagram is a large right triangle with two legs #300#ft and #x#ft and a hypotenuse #root()((300)^2+x^2)#ft by the pythagorean theorem,

#a^2+b^2=c^2#,

and another right triangle standing on top of that hypotenuse. This second, smaller triangle has one leg of #20#ft (the height of the building), and another of #root()((300)^2+x^2)#ft (because this second triangle is standing on the hypotenuse of the other, its length is the length of the hypotenuse of the first) and a hypotenuse of #d#.

From this, we know that the hypotenuse of the smaller triangle, once again making use of the pythagorean theorem, is equal to

#d=(20)^2#ft #+ (root()((300)^2+x^2))^2#ft
#d= 400#ft #+ (300)^2#ft#+x^2#ft
#d=400#ft #+ 90000#ft#+x^2#ft
#d=90400#ft #+x^2#ft.