Question

A kite needs to be flown at a height 45 feet in the air. The kite's string will be attached to a pole, at a point 5 feet above the ground. The angle of elevation from the point where the string is attached to the kite is 45º. What is the length of the rope that will hold the kite at the correct height? The length of the rope is approximately __________ feet.

Answer 1

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 =
`(opposite)/(hypotenuse)`

So, in ΔABE,

Cos(45°) =
`(AE)/(AB)`

⇒
`(√(2))/(2) = (40)/(AB)`

⇒ AB×√2 =80 (by cross multiplication)

⇒ AB =
`(80)/(√(2)) = (80√(2))/(2)` (multiplying up and down by √2)

⇒ AB = 40√2 = 56.57 (approximately)

So, the length of the rope is approximately 56.57 feet.