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geomery

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An airplane takes off from an airport.  When the airplane reaches a height of 16,500 (ft), the airplane has traveled a horizontal distance of 6000 ft, as shown in the diagram below.

Mar 22, 2021

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Part A:

tan( x )  =  opposite / adjacent

tan( x )  =  16500 / 6000

tan( x )  =  11/4

x   =   arctan( 11/4 )

x   ≈   70°

Part B:

This is the length of the third/unknown side of the triangle, which we can find using the Pythagorean Theorem. Let's call this side  "c". Then...

60002  +  165002   =   c2

$$\color{} c\ =\ \sqrt{6000^2+16500^2}$$

We could plug the above into a calculator, or we can first simplify it like this before plugging it in:

$$\color{gray} c\ =\ \sqrt{6000^2+16500^2} \\~\\ \color{gray} c\ =\ \sqrt{(4\cdot1500)^2+(11\cdot1500)^2} \\~\\ \color{gray} c\ =\ \sqrt{4^2\cdot1500^2+11^2\cdot1500^2} \\~\\ \color{gray} c\ =\ \sqrt{1500^2(4^2+11^2)} \\~\\ \color{gray} c\ =\ \sqrt{1500^2}\cdot\sqrt{(4^2+11^2)} \\~\\ \color{gray}c\ =\ 1500\sqrt{137}$$

Now if we put this into a calculator we get:

c   ≈   17557   (And this is in feet)

Mar 22, 2021