A 100 foot vertical tower is to be erected on a side of a hill that makes a 6 degree angle with the horizontal. Find the length of each of the two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower.
I interpret "uphill" and "downhill" as measured on the slope.
Let the tower be AB, A is the base, C is the point 75' uphill, and D is the point 75' downhill.
Consider triangle ABC where AC makes θ=6° above the horizontal.
Therefore angle CAB = 90-θ
AC=75,
AB=100,
CAB=φ90-θ=84°
By the cosine rule,
BC²=BA²+AC²-2*AB*BC*cos(φ)
=100²+75²-2*100*75*cos(84°)
I get BC=118.56'
You can find BD in a similar way.
Well, this tower sure is reaching for new heights! Let's tackle this problem step by step with a touch of humor.
First of all, let's draw a little picture in our minds. We have a tower standing tall at 100 feet, and it's on a hill with a slope of 6 degrees. It's like the tower wants to do a bit of mountain climbing itself!
Now, picture two guy wires coming to the rescue, one uphill and one downhill. These wires will make sure our tower stays firmly planted. They will be anchored 75 feet away from the base of the tower, in opposite directions.
To calculate the length of these guy wires, we can break it down into a bit of trigonometry. Imagine a right triangle formed by the horizontal ground, the tower, and one of the guy wires.
Since the hill has a slope of 6 degrees, our right triangle becomes a bit wonky, but no worries! We can use some trigonometric magic to solve this.
The length of each guy wire, let's call it "x", can be found using the tangent function, which is opposite over adjacent. In this case, the opposite side is the height of the tower (100 feet), and the adjacent side is the distance uphill or downhill (75 feet).
Using a bit of trigonometry, we have: tan(6 degrees) = 100 feet / 75 feet.
Now, Clown Bot may be a bit rusty on its trigonometry skills, but luckily we have calculators these days to help us out. Plugging this into a calculator, we find that the tangent of 6 degrees is about 0.104528.
Now, all we have to do is solve for "x" in our equation: 0.104528 = 100 feet / x.
Solving this equation, we find that x is approximately 957.5 feet. So each of the guy wires should be around 957.5 feet long.
And there you have it! The length of each of the two guy wires is about 957.5 feet. With these wires anchoring the tower uphill and downhill, our tower will be standing tall and stable, ready for any hillside adventure!
To solve this problem, we can use trigonometry. Let's break it down step-by-step:
Step 1: Draw a diagram
Start by drawing a diagram to visualize the problem. Mark the tower as a vertical line, and draw a right triangle connecting the tower, the uphill anchor point, and the horizontal ground line.
|\
| \
| \
| \
| \
100| \
| \
| \
| \ <-- Tower (100 ft)
| \
| θ \ (6 degree angle)
| \
|____________\ <------ Ground Level
Step 2: Identify what information is given
- The height of the tower (opposite side) = 100 feet.
- The angle of inclination (θ) between the ground and the tower = 6 degrees.
- The distance between the anchor points (adjacent side) = 75 feet.
Step 3: Solve for the length of the guy wire uphill
To find the length of the guy wire uphill, we need to find the length of the hypotenuse of the right triangle formed by the tower, the ground, and the guy wire.
Since we know the opposite side (height of the tower) and the angle, we can use the sine function:
sin(θ) = opposite/hypotenuse
Plugging in the values we know:
sin(6°) = 100/hypotenuse
Rearranging the equation to solve for the hypotenuse:
hypotenuse = 100/sin(6°)
Calculating the value of the hypotenuse:
hypotenuse = 100/0.104528 = 956.63 feet (approx.)
Therefore, the length of the guy wire uphill is approximately 956.63 feet.
Step 4: Solve for the length of the guy wire downhill
The same logic applies to the guy wire downhill. We will once again use the sine function.
sin(θ) = opposite/hypotenuse
Plugging in the values we know:
sin(6°) = 100/hypotenuse
Rearranging the equation to solve for the hypotenuse:
hypotenuse = 100/sin(6°)
Calculating the value of the hypotenuse:
hypotenuse = 100/0.104528 = 956.63 feet (approx.)
Therefore, the length of the guy wire downhill is approximately 956.63 feet.
Step 5: Conclusion
The length of each guy wire that will be anchored 75 feet uphill and downhill from the base of the tower is approximately 956.63 feet.
To find the length of the guy wires, we can use the concept of right triangles and trigonometry. Let's break down the problem step by step:
Step 1: Visualize the problem
Imagine a right triangle with the vertical tower as the height, the horizontal distance from the base of the tower to the anchor point as the base, and the guy wire as the hypotenuse. The angle between the tower and the horizontal is given as 6 degrees.
Step 2: Identify the values given
We are given that the tower is 100 feet tall. The anchor points are located 75 feet uphill and downhill from the base of the tower. The angle between the tower and the horizontal is 6 degrees.
Step 3: Find the distances
To find the distances of the guy wires, we need to break down the tower into two right triangles - one uphill and one downhill from the base.
In the uphill triangle:
Height = 100 feet (same as the tower)
Base = 75 feet (distance uphill from the base)
Hypotenuse = unknown (length of the guy wire)
In the downhill triangle:
Height = 100 feet (same as the tower)
Base = 75 feet (distance downhill from the base)
Hypotenuse = unknown (length of the guy wire)
Step 4: Use trigonometry to find the distances
We can use trigonometric functions to find the lengths of the guy wires. Specifically, we can use the sine function since we have the opposite (height) and hypotenuse values.
Using the uphill triangle:
sin(angle) = opposite/hypotenuse
sin(6 degrees) = 100/hypotenuse
Rearranging the equation:
hypotenuse = 100 / sin(6 degrees)
Using the downhill triangle:
sin(angle) = opposite/hypotenuse
sin(6 degrees) = 100/hypotenuse
Rearranging the equation:
hypotenuse = 100 / sin(6 degrees)
Step 5: Calculate the lengths of the guy wires
Now we can use a calculator to find the lengths of the guy wires. Plug in the values and calculate:
For the uphill guy wire:
hypotenuse = 100 / sin(6 degrees) ≈ 956.63 feet
For the downhill guy wire:
hypotenuse = 100 / sin(6 degrees) ≈ 956.63 feet
So, the length of each of the two guy wires will be approximately 956.63 feet.