A smokestack is 150
feet high. A guy wire must be fastened to the stack
40
feet from the top. The guy wire makes an angle of 40∘
with the ground. Find the length of the guy wire rounded to the nearest foot.
- I've got no idea where to start or what to do. I understand that it is 150 high, but do we add on the other 40 to that? It seems that there are two equations going on, overall I am just very confused. Please and thank you for any help.
Well, let's start by visualizing the situation. Imagine a right-angled triangle, where the smokestack is the vertical side, the guy wire is the hypotenuse, and the distance 40 feet from the top is the base.
Now, we can use trigonometry to solve this problem. Since we have the angle and the length of the adjacent side, we can use the cosine function to find the length of the guy wire.
The cosine of an angle is equal to the adjacent side divided by the hypotenuse length. So in this case, cos(40°) = 40 feet / hypotenuse length.
Now we can rearrange this equation to solve for the hypotenuse length:
hypotenuse length = adjacent side / cos(40°)
Plugging in the values, the hypotenuse length would be approximately equal to 40 feet / cos(40°).
So, all you need to do is find the cosine of 40° and divide 40 feet by that value. Rounded to the nearest foot, that should give you the length of the guy wire. Good luck!
To solve this problem, we can use trigonometry and the concept of right triangles. Here's what we need to do step by step:
Step 1: Visualize the problem
Draw a diagram to accurately represent the situation. Draw a vertical line to represent the smokestack, and label it as 150 feet. Then draw a horizontal line from the 40 feet mark to the ground. This horizontal line represents the guy wire.
Step 2: Identify the right triangle
In the diagram, we have a right triangle formed by the height of the smokestack, the distance from the top of the smokestack to where the guy wire is fastened, and the guy wire itself. The angle between the ground and the guy wire is also given as 40 degrees.
Step 3: Understand trigonometric ratios
We can use trigonometric ratios to solve the problem. In this case, the most useful ratio is the tangent ratio (tan). The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, we can use the tangent to find the length of the guy wire.
Step 4: Apply the tangent ratio
Let's define the variables:
- h: height of the smokestack (150 ft)
- x: distance from the top of the smokestack to where the guy wire is fastened (40 ft)
- θ: angle between the ground and the guy wire (40°)
- g: length of the guy wire (what we need to find)
According to the tangent ratio, tan(θ) = h / x
Substituting the values, we have tan(40°) = 150 / x
To solve for x, we rearrange the equation as follows:
x = 150 / tan(40°)
Step 5: Solve for x
Using a calculator, compute the value of x:
x ≈ 150 / 0.8391 ≈ 178.69 ft
Step 6: Find the length of the guy wire
Now that we have x, we can find the length of the guy wire using the Pythagorean theorem. In the right triangle, the guy wire is the hypotenuse, so we can use the formula:
g = √(h² + x²)
Substituting the values:
g = √(150² + 178.69²)
g ≈ √(22500 + 31881.96)
g ≈ √54381.96
g ≈ 233.16 ft
Rounded to the nearest foot, the length of the guy wire is approximately 233 feet.
this is trig
the wire is fastened 40 feet BELOW the top ... how high up?
the wire is the hypotenuse of a right triangle
... you know the length of one side (height of attachment point)
... you know the angle with the ground
draw a sketch
Read carefully, the wire is attached 40 ft from the top, so the wire is attached 110 ft above the ground.
So now it becomes very simple, a right-angled triangle with base angle of 40°, and an opposite side of 110 ft.
We want the hypotenuse h, so
sin40° = 110/h
solve for h, see how easy it becomes ?
(let me know what you get)