Two guy wires for the radio tower of SPI station made 58° and 49° angles with the horizontal as shown in the figure. If the ground anchors for each wire were 150 ft apart and assuming that the wires are straight, find the length of the shorter wire
Of course we can't show figures on the website, but I am assuming
this question is of the standard type.
Assume that the guy wires are attached at the same height up the tower.
Call the points where the wires are anchored on the ground as A and B
so we know AB = 150 ft.
Let the base of the tower be C and its top as D
Using standard geometry, angle ABD = 180-58 or 122°,
then angle ADB = 9°
The shorter wire would be DB
By the sine law:
150/sin9° = DB/sin49°
DB = 150sin49/sin9 = 723.67 ft
Normally this question continues to find the height of the tower or more
interesting facts
No figure shown, so there's something left undetermined.
Suppose we label things as follows:
T = top of pole
P = bottom of pole
so the pole's height h = PT
A = wire 1 anchor point (58°)
B = wire 2 anchor point (49°)
x = PA
y = PB
z = AB = 150
Then, if ∡APB is θ, we have
x^2 + y^2 - 2xy cosθ = 150
h/x = tan58°
h/y = tan49°
Since we don't know θ, we can't solve this yet. So, assuming the wires are anchored on opposite sides of the pole, we have
h cot58° + h cot49° = 150
h = 100.39
so the length q of the shorter wire is
h/q = sin58°
q = 118.38
To find the length of the shorter wire, we can use the concept of trigonometry and the given angles.
Let's break down the problem and label the important information:
- We have two guy wires (assumed to be straight) for the radio tower at SPI station.
- The ground anchors for each wire are 150 ft apart.
- One guy wire makes a 58° angle with the horizontal.
- The other guy wire makes a 49° angle with the horizontal.
Now, let's consider the right triangles formed by each guy wire. We can use the tangent function to relate the angles to the lengths of the wires.
For the first guy wire (with 58° angle):
- Let's assume the length of this wire is 'x' ft.
- The opposite side of the triangle is the vertical distance, which will be x * tan(58°).
For the second guy wire (with 49° angle):
- Let's assume the length of this wire is 'y' ft.
- The opposite side of the triangle is the vertical distance, which will be y * tan(49°).
Since the ground anchors for each wire are 150 ft apart, we can write the following equation:
y = x + 150
Now, we can set up a system of equations using the information we derived:
1. x * tan(58°) = y * tan(49°)
2. y = x + 150
Solving this system of equations will give us the values of x and y, representing the lengths of the two guy wires.
You can use a scientific calculator or an online trigonometric calculator to calculate the tangent of an angle and solve the equations to find the length of the shorter wire (x).