A 100 foot tall antenna sits part way up a hill. The hill makes an angle to 12 degrees with the horizontal. In other words, if you were going to walk up the hill, you would walk at an angle of 12 degrees. To keep the antenna stable, it must be anchored by 2 cables.The distance from the base of the antenna to the down point DOWN hill is 95 feet.Ignore the amount of cable needed to fasten the cable to the antenna or to the tie downs. How much cable is needed?
Please Draw and label if you can or give me the idea of this question.
I assume you only want the length of the cable going down the hill.
Make a sketch.
On mine I have a triangle with sides 100 and 95 with an angle of 102 degrees between them.
The cosine law is just screaming at you here.
x^2 = 100^2 + 95^2 - 2(100)(95)cos102
= ....
you do the button-pushing.
careful with the last term, since cos102 is negative, you will be adding it.
To solve this problem, we can start by drawing a diagram to visualize the situation.
1. Draw a horizontal line to represent the ground.
2. From one end of the ground line, draw a vertical line upwards to represent the 100-foot tall antenna.
3. From the base of the antenna, draw a diagonal line downward towards the ground to represent the hill, making a 12-degree angle with the horizontal.
The diagonal line should intersect the ground line at some point. Let's label that point P, which is the down point down the hill, and the distance from the base of the antenna to point P is given as 95 feet.
Now, we need to find the length of the cables needed to anchor the antenna. Let's assume the two cables are equally in length.
To solve for the length of one cable, we can use trigonometry. In this case, we can use the sine function since we have the opposite side (95 feet) and the hypotenuse (length of the cable).
The equation for this will be:
sin(12 degrees) = Opposite / Hypotenuse
Using this equation, we can solve for the length of one cable:
sin(12 degrees) = 95 feet / Hypotenuse
To solve for the length of one cable, we rearrange the equation:
Hypotenuse = 95 feet / sin(12 degrees)
Using a calculator, we can find the value of sin(12 degrees) ≈ 0.2079
Now, we can substitute this value into the equation:
Hypotenuse = 95 feet / 0.2079
Calculating this, we find:
Hypotenuse ≈ 456.364 feet
Since we assumed the two cables are equally in length, the total length of the cables needed is:
Total cable length = 2 * Hypotenuse
Total cable length ≈ 2 * 456.364 feet
Total cable length ≈ 912.728 feet
Therefore, approximately 912.728 feet of cable is needed to anchor the antenna.