A kite flying at a height of 60 metres above the ground is attached to string that makes an angle of 45° with the ground. What is the length (in metres) of the string?
60√2 metres
The kite's height above the ground, the string, and the horizontal ground form a right triangle, with the string as the hypotenuse and the height as the side opposite the \(45\degree\) angle at the ground.
By definition, \(\sin 45\degree = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{\text{height}}{\text{string}}\).
So string \(= \dfrac{\text{height}}{\sin 45\degree}\).
Substitute height \(= 60\ \text{m}\) and \(\sin 45\degree = \dfrac{1}{\sqrt{2}}\): string \(= \dfrac{60}{1/\sqrt{2}} = 60 \times \sqrt{2} = 60\sqrt{2}\) m.
As a check, \(60\sqrt{2} \approx 84.85\ \text{m}\), which is sensibly longer than the 60 m height.
Key concept: the angle of elevation of the string links the known vertical height to the unknown hypotenuse through the sine ratio.
Hence the length of the string is 60√2 metres.
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