Question

A kite has 120m of string attached to it, when at an elevation of 60 degree. How far is it above the hand holding it?

Answer 1

The elevation is `103.92 or 60sqrt(3)` meters

Explanation:

The are two ways of approaching this. One is using the sine relationship, and the other is using a triangular identity.

The identity I'm talking about is the side ratios of a 30-60-90 triangle.

The short side (from the 60-degree to the 90) is ratio of 1.

The hypotenuse is ratio of 2

The long side (from the 90 to the 30-degree) is ratio of `sqrt(3)`

So, if the string length is 120, and is forming a 60-degree angle with the horizontal, the ratio of the hypotenuse to the altitude is:

`2:sqrt(3)`

Using this:

`"hyp"=2/sqrt(3)"alt" rArr 120=2/sqrt(3)"alt"`

`sqrt(3)/2xx120="alt"`

`color(red)("alt"=60sqrt(3)=103.92`

The other method is using sine. Since the side relationship of sine is:

`Sin(x)="opposite"/"hypotenuse"`

we can re-write this to get a direct relationship, knowing that the 'opposite' side is the altitude:

`Sin(60)="alt"/120 rArr 120xxSin(60)="alt"`

It turns out that `Sin(60)=sqrt(3)/2~=0.866`, which bolsters the triangle identity above... finishing it up:

`120sqrt(3)/2="alt"`

`color(blue)("alt"=60sqrt(3)=103.92)`