A smokestack is

200 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.

200 - 30 = 170

sin 42 = 170/x

Well, if I were a guy wire, I would want to be long enough to reach the top of the smokestack, but not so long that I end up tripping over myself. Let's see what we can do here.

First, let's pretend that the smokestack is a really tall clown hat (I have a thing for clown-related imagery, sorry). We can think of the guy wire as a string that goes from the top of the clown hat to a point on the ground. Now, all I have to do is use a little bit of trigonogiggles to find the length of that string.

Using some trigonometry, we can determine that the length of the guy wire is given by the equation:

Length of guy wire = height of smokestack / sin(angle with the ground)

Plugging in the given values, we have:

Length of guy wire = 200 ft / sin(42.0°)

Now, I won't bore you with the math (or at least I'll try not to), but after doing some calculations, we get:

Length of guy wire ≈ 311.74 ft

So, the length of the guy wire is approximately 311.74 feet. That should be long enough to keep the smokestack upright and well-balanced, just like a clown on a tall unicycle!

To find the length of the guy wire, we can use trigonometry. Let's call the length of the guy wire "x".

We can create a right triangle with the height of the smokestack (200 ft), the distance from the top to where the guy wire is fastened (30.0 ft), and the length of the guy wire (x). The angle between the ground and the guy wire is given as 42.0°.

Using the trigonometric ratio tangent (tan), we can set up the equation:

tan(42.0°) = (200 ft - 30.0 ft) / x

Now, let's solve for x.

x * tan(42.0°) = (200 ft - 30.0 ft)

x = (200 ft - 30.0 ft) / tan(42.0°)

x ≈ 168.62 ft

Therefore, the length of the guy wire is approximately 168.62 feet.

To find the length of the guy wire, we can use trigonometry.

Let's take the height of the smokestack as the vertical side of a right triangle, and the distance where the guy wire is fastened as the horizontal side of the triangle. The guy wire forms an angle of 42.0° with the ground, which is the angle opposite to the vertical side.

Using the sine function, we can relate the length of the guy wire to the height and the angle:

sin(angle) = opposite/hypotenuse

In this case, the opposite side is the height of the smokestack, and the hypotenuse is the length of the guy wire.

Given that the height of the smokestack is 200 ft and the angle is 42.0°, we can substitute these values into the equation:

sin(42.0°) = 200/hypotenuse

Now, we can solve for the length of the guy wire (hypotenuse):

hypotenuse = 200 / sin(42.0°)

Calculating this value:

hypotenuse ≈ 200 / 0.6691 ≈ 298.87 ft

Therefore, the length of the guy wire is approximately 298.87 ft.