A radio transmission tower is 170 feet tall. How long should a guy wire be if it is to be attached 7 feet from the top and is to make an angle of 30° with the ground? Give your answer to the nearest tenth of a foot

A radio transmission tower is 170 feet tall. How long should a guy wire be if it is to be attached 7 feet from the top and is to make an angle of 30° with the ground? Give your answer to the nearest tenth of a foot

To find the length of the guy wire, we can use trigonometry. We know the height of the tower (170 feet) and the distance from the top to where the guy wire is attached (7 feet). We can use the tangent function to calculate the length of the guy wire.

First, let's find the opposite side to the angle of 30°:
Opposite side = tangent(angle) × adjacent side
Opposite side = tan(30°) × 7

Using a scientific calculator or trigonometric table, tan(30°) is approximately 0.5774.

Opposite side = 0.5774 × 7
Opposite side ≈ 4.0418 feet

Next, we need to find the hypotenuse of the right triangle formed by the tower, the guy wire, and the ground. The hypotenuse is the length of the guy wire.
Hypotenuse = height of the tower + opposite side
Hypotenuse = 170 + 4.0418

Hypotenuse ≈ 174.0418 feet

So, the length of the guy wire should be approximately 174.0 feet (rounded to the nearest tenth of a foot).

To find the length of the guy wire, we can use trigonometric functions. In this case, we can use the sine function.

The sine function is defined as the ratio of the length of the side opposite the angle to the hypotenuse. In this scenario, the guy wire acts as the hypotenuse, and the side opposite the angle is the height of the tower minus the distance at which it is attached.

Let's break down the problem and solve it step by step:

1. Start by calculating the length of the side opposite the angle:

Side opposite the angle = height of the tower - distance attached
= 170 ft - 7 ft
= 163 ft

2. Now, we can use the sine function to find the length of the guy wire:

sine(angle) = opposite / hypotenuse

Rearranging the formula, we have:

hypotenuse = opposite / sine(angle)

hypotenuse = 163 ft / sine(30°)

Using a calculator, we can find the numerical value of sine(30°) ≈ 0.5:

hypotenuse = 163 ft / 0.5
= 326 ft

3. Finally, the length of the guy wire rounded to the nearest tenth of a foot is 326 feet.

So, the answer is: The guy wire should be approximately 326 feet long.

Draw a rt triangle:

X = Hor. side.
Y = 7 Ft = Ver. side.
Z = Hyp. = Length of guy wire.
A = 30 Deg.

Z = Y/sinA = 7 / sin30 = 14 Ft.


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