A communications tower has many guy wires supporting it. Two of these guy wires are 10.0 m and 8.0 long. They are attached at the same point on the ground. The longer wire has an angle of inclination of 60 degrees.
a) How far from the base of the tower are the wires attached to the ground? (Which side am I looking for? Do I use Cos?)
b) What is the angle of inclination of the shorter guy wire?
c) How far apart are the points where the guy wires are attached to the tower?
Give measures to the nearest tenth.
yes, use cos to find how far away the wires are anchored.
Then use that distance and cos again to find the smaller angle.
Then use sin to find the heights on the tower. Or, equivalently, use the Pythagorean Theorem to find the heights, since you have the base and tbhe hypotenuses.
To solve this problem, we can use trigonometric functions to find the lengths and angles involved.
a) To find the distance from the base of the tower to where the wires are attached to the ground, we need to find the horizontal component (adjacent side) of the longer wire using cosine (cos).
Using the formula: adjacent = hypotenuse * cos(angle)
where:
hypotenuse = length of the longer wire = 10.0 m
angle = angle of inclination of the longer wire = 60 degrees
adjacent = 10.0 m * cos(60 degrees)
adjacent = 10.0 m * 0.5
adjacent = 5.0 m
Therefore, the distance from the base of the tower to where the wires are attached to the ground is 5.0 meters.
b) To find the angle of inclination of the shorter guy wire, we can use the concept of complementary angles. The angle between the horizontal and the longer wire is 60 degrees. The other angle will be 90 degrees (as a total of 180 degrees right angle).
Since the shorter wire is attached to the same point on the ground as the longer wire, the angle of inclination of the shorter wire will be the complementary angle to the angle of inclination of the longer wire.
angle of inclination of the shorter wire = 90 degrees - angle of inclination of the longer wire
angle of inclination of the shorter wire = 90 degrees - 60 degrees
angle of inclination of the shorter wire = 30 degrees
Therefore, the angle of inclination of the shorter guy wire is 30 degrees.
c) To find the distance between the points where the guy wires are attached to the tower, we need to find the vertical component (opposite side) of the longer wire using sine (sin).
Using the formula: opposite = hypotenuse * sin(angle)
where:
hypotenuse = length of the longer wire = 10.0 m
angle = angle of inclination of the longer wire = 60 degrees
opposite = 10.0 m * sin(60 degrees)
opposite = 10.0 m * 0.866 (rounded to three decimal places)
opposite = 8.66 m (rounded to two decimal places)
Therefore, the distance between the points where the guy wires are attached to the tower is approximately 8.7 meters.
a) To find the distance from the base of the tower to where the wires are attached to the ground, we can use the cosine function, as you correctly mentioned. We are looking for the side adjacent to the given angle, so we will use the cosine function.
Let's label the longer wire as "L" and the shorter wire as "S". The longer wire has an angle of inclination of 60 degrees, so we will use the cosine of 60 degrees to find the length of the side adjacent to it.
Using the cosine function:
cos(60) = Adjacent / Hypotenuse
cos(60) = x / 10.0 (where x is the distance from the base of the tower to where the wire is attached to the ground)
Rearranging the equation to solve for x:
x = cos(60) * 10.0
Using a calculator, we find:
x ≈ 5.0
Therefore, the distance from the base of the tower to where the longer wire is attached to the ground is approximately 5.0 meters.
b) To find the angle of inclination of the shorter guy wire, we need to use the inverse cosine function. We are given the length of the shorter wire (8.0 m) and we are looking for the angle opposite to it.
Using the inverse cosine function:
cos^(-1)(Adjacent / Hypotenuse) = Angle
Using the given values:
cos^(-1)(8.0 / 10.0) = Angle
Using a calculator, we find:
Angle ≈ 38.7 degrees
Therefore, the angle of inclination of the shorter guy wire is approximately 38.7 degrees.
c) To find the distance between the points where the guy wires are attached to the tower, we can subtract the distance of the point of attachment of the shorter wire from the distance of the point of attachment of the longer wire.
Using the values we have:
Distance between attachment points = Length of longer wire - Length of shorter wire
Distance between attachment points = 10.0 - 8.0
Therefore, the distance between the points where the guy wires are attached to the tower is 2.0 meters.