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 = [tex] \frac{opposite}{hypotenuse} [/tex]
So, in ΔABE,
Cos(45°) = [tex] \frac{AE}{AB} [/tex]
⇒ [tex] \frac{\sqrt{2}}{2} = \frac{40}{AB} [/tex]
⇒ AB×√2 =80 (by cross multiplication)
⇒ AB = [tex] \frac{80}{\sqrt{2}} = \frac{80\sqrt{2}}{2} [/tex] (multiplying up and down by √2)
⇒ AB = 40√2 = 56.57 (approximately)
So, the length of the rope is approximately 56.57 feet.
