A vertical radio mast is supported by 3 equal
wires.One end of each wire is fixed to the top
of the mast and the other ends form an
equilateral triangle on horizontal ground.If the
length of each wire is twice the height of the
mast,find the angle btw any two adjacent wire?
Show solution ste
Draw a diagram. If we let
T be the top of the mast
O be the base of the mast
P,Q be attachment points of two wires (PQ forms one side of the base of the triangle on the ground)
M be the midpoint of PQ
OV = 1
PV = 2
so, OP = √3
triangle MOP is a 30-60-90 right triangle, so
OP = √3
OM = √3/2
MP = 3/2
in right triangle MVP,
MP = 3/2
PV = 2
We want the angle PVQ=θ, and
sin θ/2 = MP/PV = 3/4
Tis the top of the mast
To solve this problem, we can start by drawing a diagram to visualize the given information. Let's represent the vertical radio mast as a straight line and the three equal wires as three lines connecting the top of the mast to points on the ground forming an equilateral triangle.
Let's label the height of the mast as "h" and the length of each wire as "2h" since it is twice the height of the mast.
Now, let's consider one of the wires and draw it on the diagram. We can label this wire as "W1". The other two wires can be labeled as "W2" and "W3".
Since all three wires are equal in length, we can consider the triangle created by connecting the other endpoints of W2 and W3 as an isosceles triangle. Let's label this triangle as "ABC", with A being the top vertex of the equilateral triangle.
To find the angle between any two adjacent wires, we can find the measure of one of the angles of the triangle ABC.
Since triangle ABC is isosceles, we can conclude that angle ABC is equal to angle ACB.
Let's use properties of an equilateral triangle to find the measure of angle ABC.
In an equilateral triangle, all angles are equal and measure 60 degrees.
Therefore, angle ABC measures 60 degrees.
Since angle ABC is the same as angle ACB, we can conclude that the angle between any two adjacent wires is 60 degrees.
To find the angle between any two adjacent wires, we can draw a diagram and use some trigonometry.
Let's assume the height of the mast is h. Since each wire is twice the height of the mast, the length of each wire will be 2h.
Now, let's draw the diagram. We have a vertical mast, and the three wires are attached to the top of the mast and form an equilateral triangle on the ground.
Since the wires are equilateral, we know that all the angles in the triangle are 60 degrees.
We want to find the angle between any two adjacent wires. Let's call this angle x.
Since the wires are attached to the top of the mast, we have a right triangle formed between the mast and one of the wires. The angle between the mast and the wire is 90 degrees.
Let's consider one of the wires. We have a right triangle with the height of the mast (h), the length of the wire (2h), and the angle between the mast and the wire (90 degrees).
Using trigonometry, we can use the cosine function to find the angle between the wire and the ground:
cos(x) = h / (2h)
cos(x) = 1/2
To find the angle x, we can take the inverse cosine (arccos) of both sides:
x = arccos(1/2)
x = 60 degrees
So, the angle between any two adjacent wires is 60 degrees.