Questions: A radio transmission tower is 140 feet tall. How long should a guy wire be if it is to be attached 8 feet from the top and is to make an angle of 25° with the ground? Give your answer to the nearest tenth of a foot.

Solution Steps

To solve this problem, we can use trigonometry. The guy wire, the tower, and the ground form a right triangle. The height from where the wire is attached to the ground is 132 feet (140 - 8). We can use the sine function, where the sine of the angle is equal to the opposite side (height from attachment to ground) divided by the hypotenuse (length of the wire). We can rearrange this to solve for the hypotenuse.

The radio transmission tower is 140 feet tall, and the guy wire is attached 8 feet from the top. Therefore, the height from the attachment point to the ground is: \[ \text{Height} = 140 - 8 = 132 \text{ feet} \]

The angle between the guy wire and the ground is given as \(25^\circ\). To use trigonometric functions, we need to convert this angle to radians: \[ \text{Angle in radians} = \frac{25 \times \pi}{180} \approx 0.4363 \text{ radians} \]

In the right triangle formed by the tower, the guy wire, and the ground, the sine of the angle is the ratio of the opposite side (height from the attachment point to the ground) to the hypotenuse (length of the guy wire). Thus, we have: \[ \sin(25^\circ) = \frac{132}{\text{Guy wire length}} \] Solving for the guy wire length: \[ \text{Guy wire length} = \frac{132}{\sin(25^\circ)} \approx 312.3 \text{ feet} \]

Final Answer