A 12-foot guy wire is to be attached to a tree at a 30 degree angle to the ground. At what height on the tree should the guy wire be attached? How far from the tree should the wire be attached to the ground?
a. We form a rt triangle:
A = 30 Deg. = Angle between hyp. and
hor. side.
Y = Ht. of guy wire attached to tree.
Z = hyp. = 12 Ft guy wire.
Y = 12*sin30 = 6 Ft.
b. X = 12*cos30 = 10.4 Ft.
To find the height at which the guy wire should be attached to the tree, we can use trigonometry. We have a right triangle where the guy wire is the hypotenuse, the height on the tree is the opposite side, and the distance from the tree to the attachment point on the ground is the adjacent side.
Let's start by finding the height on the tree. We can use the sine function to solve for the opposite side:
sin(angle) = opposite / hypotenuse
sin(30 degrees) = height / 12 feet
To find the height, we can multiply both sides of the equation by 12 feet:
height = 12 feet * sin(30 degrees)
Using a calculator, we find:
height = 12 feet * 0.5
height = 6 feet
So, the guy wire should be attached to the tree at a height of 6 feet.
Next, let's find the distance from the tree to the attachment point on the ground. We can use the cosine function to solve for the adjacent side:
cos(angle) = adjacent / hypotenuse
cos(30 degrees) = distance / 12 feet
To find the distance, we can multiply both sides of the equation by 12 feet:
distance = 12 feet * cos(30 degrees)
Using a calculator, we find:
distance = 12 feet * 0.866
distance ≈ 10.392 feet
So, the wire should be attached to the ground at a distance of approximately 10.392 feet from the tree.