a smokestack is 190 feet high. a guy wire must be fastened to the stack 30.0 feet from the top. the guy wire makes an angle of 39.0deg with the ground. find the length of the guy wire
as always, draw a diagram. You can easily see that the length w is
(190-30)/w = sin 39°
A smokestack is 180 ft high. A guy wire must be fastened to the stack 30.0 ft from the top. The guy wire makes an angle of 42.0° with the ground. Find the length of the guy wire?
To find the length of the guy wire, we can use trigonometry.
Let's label the height of the smokestack as "h" and the length of the guy wire as "g". We are given that the smokestack is 190 feet high, and the guy wire is fastened to the stack 30.0 feet from the top. This means the remaining height of the smokestack is 190 - 30 = 160 feet.
We are also given that the guy wire makes an angle of 39.0 degrees with the ground.
Using trigonometry, we can use the sine function to relate the angle, the opposite side (h), and the hypotenuse (g) of the right triangle formed by the smokestack and the guy wire:
sin(angle) = opposite / hypotenuse
sin(39.0) = h / g
Rearranging the equation, we can solve for g:
g = h / sin(angle)
g = 160 / sin(39.0)
Using a calculator, we can find that sin(39.0) is approximately 0.6293. Plugging in this value, we have:
g = 160 / 0.6293
Simplifying this further, we find the length of the guy wire:
g ≈ 254.41 feet
Therefore, the length of the guy wire is approximately 254.41 feet.
To find the length of the guy wire, we can use trigonometry.
First, we can visualize the situation. Imagine a right triangle formed by the smokestack, the guy wire, and the ground:
/|
/ |
L / |
/ θ | |
/__________|
In this triangle, the height of the smokestack from the ground is the adjacent side (A), and the distance between the attachment point (30.0 ft) and the top of the smokestack is the opposite side (O). We are given the angle (θ) that the guy wire makes with the ground.
We can use the tangent function to find the length of the guy wire. The tangent function relates the opposite side and the adjacent side of a right triangle:
tan(θ) = opposite/adjacent
In this case, we want to solve for the opposite side, which is the length of the guy wire (L), and we know the adjacent side (A) is 30.0 ft and the angle (θ) is 39.0 degrees.
So, we can rearrange the equation to solve for L:
L = opposite = tan(θ) * adjacent
L = tan(39.0°) * 30.0 ft
Now we can calculate the length of the guy wire:
L = tan(39.0°) * 30.0 ft ≈ 23.68 ft
Therefore, the length of the guy wire is approximately 23.68 feet.