Question

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

Answer 1

`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.