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.
To solve this problem, let's label the different points and lengths:
- Let A be the top of the antenna
- Let B be the base of the antenna
- Let C be the point where the cable is anchored down the hill
- Let D be the point where the cable is anchored up the hill
Now let's draw a diagram to understand the situation:
```
A (Top of antenna)
/|
/ |
/ |
/ | 100 ft (height of antenna)
/ θ |
B____C__D (anchor points on the hill)
95 ft
```
In this diagram, angle θ represents the angle of the hill from the horizontal and is given as 12 degrees. The height of the antenna is 100 feet, and the distance from the base of the antenna to the down point downhill is 95 feet.
To find the length of the cable needed, we can use trigonometry. Since the cable forms a right triangle with the hill, we can use the sine function to determine the length of the cable.
The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side (BC) is the height of the antenna (100 feet), and the hypotenuse (BD) is the length of the cable.
Using the sine function:
sin(θ) = BC / BD
We know that sin(θ) = sin(12°) and BC = 100 ft, so we can rearrange the equation to solve for BD:
BD = BC / sin(θ)
Substituting the given values:
BD = 100 ft / sin(12°)
Using a calculator:
BD ≈ 475.68 ft (rounded to two decimal places)
Therefore, the length of the cable needed to anchor the antenna is approximately 475.68 feet.
To solve this question, we need to understand the given information and draw a diagram. Let's break down the problem step by step:
1. Draw a horizontal line to represent the ground level. Label this line as the "horizontal."
horizontal
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2. Draw a vertical line standing on the ground to represent the antenna. Label this line as the "antenna."
horizontal
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antenna
3. Draw a line from the base of the antenna to the down point downhill at an angle of 12 degrees with the horizontal. Label this line as "hill."
horizontal
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| hill
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antenna
4. Label the distance from the base of the antenna to the downhill point as 95 feet.
horizontal
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| 95 ft
| hill
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antenna
5. We need to find the length of the cables that anchor the antenna to the ground. Let's label this length as "cable."
horizontal
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| 95 ft
| hill
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cable
antenna
Now that we have the diagram, we can proceed to find the length of the cable needed.
To find the length of the cable, we need to consider the right triangle formed by the hill, the horizontal, and the cable.
In this right triangle:
- The length of the horizontal line is 95 ft.
- The angle between the horizontal line and the hill is 12 degrees.
Using trigonometric functions, we can find the length of the cable. In particular, we will use the sine function.
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In our case, the hill is opposite to the angle, and the cable is the hypotenuse.
We can use the formula: sin(angle) = opposite/hypotenuse
sin(12 degrees) = hill/cable
Rearranging the formula to solve for the cable, we get:
cable = hill / sin(12 degrees)
Substituting the given values, we have:
cable = 95 ft / sin(12 degrees)
Using a scientific calculator, find the sine of 12 degrees, and then divide 95 feet by that value to find the length of the cable.
Therefore, the cable needed to anchor the antenna is the result of this calculation.