The length of the rope is approximately 56.57 feet.
Here, Kite's position is at point A which is 45 ft above the ground.
So, in the diagram AC= 45 ft.
B is the point on the pole above 5 ft. from ground, where the kite's string will be attached. So, BD= 5 ft
In the diagram, we will draw a line from point B parallel to the ground which will meet the line AC at point E.
As, BD= 5 ft, so EC= 5 ft also. Now, AE= AC - EC = 45- 5 = 40 ft.
The angle of elevation of the string from the kite's position is 45°
For that, ∠ABE = 45° also (according to the Alternate Interior Angles)
So, in right angle triangle ABE,
in respect of ∠ABE, opposite side(AE)= 40 and we need to find the length of the rope, which is hypotenuse AB.
As, Cosθ = ![\frac{opposite}{hypotenuse}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bopposite%7D%7Bhypotenuse%7D%20%20)
So, in ΔABE,
Cos(45°) = ![\frac{AE}{AB}](https://tex.z-dn.net/?f=%20%5Cfrac%7BAE%7D%7BAB%7D%20)
⇒ ![\frac{\sqrt{2}}{2} = \frac{40}{AB}](https://tex.z-dn.net/?f=%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%20%3D%20%5Cfrac%7B40%7D%7BAB%7D%20)
⇒ AB×√2 =80 (by cross multiplication)
⇒ AB =
(multiplying up and down by √2)
⇒ AB = 40√2 = 56.57 (approximately)
So, the length of the rope is approximately 56.57 feet.