To secure a 610-meter radio tower against high winds, guy wires are attached to a ring on the tower. The ring is 7 meters from the top. The wires form a 69o angle with the ground. To the nearest hundredth meter find the length of the guy wire

Make a sketch to see that

sin 69° = 603/L
L = 603/sin69 = appr 562.95 m

To find the length of the guy wire, we can use trigonometry and the given information about the angle and distance from the top.

Let's assume the length of the guy wire is "x" meters.

Since the guy wires form a 69° angle with the ground, we can consider this angle as one of the acute angles in a right triangle. The guy wire serves as the hypotenuse of this right triangle.

The side adjacent to the angle of 69° is the distance from the top of the tower to the ring, which is 7 meters.

Using trigonometric functions, specifically the cosine function, we can express the relationship between the adjacent side, hypotenuse, and the angle:

cos(69°) = adjacent / hypotenuse

cos(69°) = 7 / x

To find the length of the guy wire, we need to isolate "x". We can do this by rearranging the equation:

x = 7 / cos(69°)

Using a calculator, we can evaluate the cosine of 69° and divide 7 by that value to find the length of the guy wire:

x = 7 / cos(69°) ≈ 23.84 meters

Therefore, the length of the guy wire, to the nearest hundredth meter, is approximately 23.84 meters.