A pole tilts 12 degrees from the vertical, away from the sun, casts a 34 foot long shadow on level ground. The angle of elevation from the end of the shadow to the top of the pole is 64 degrees. How long is the pole?
Using the Law of Sines,
L / sin64 = 34 / sin(90-12),
L / sin64 = 34 / sin78,
Multiply both sides by sin64:
L = 34sin64 / sin78 = 31.2ft = Length
of pole.
tan x=h/¡Ì3h
tan x=1/¡Ì3
tan x=30¡ã
the angle of elevation is 30¡ã
To find the length of the pole, we can use trigonometry. Let's denote the length of the pole as "x".
First, let's find the height of the pole using the angle of elevation. The angle of elevation is the angle between the ground and the line from the end of the shadow to the top of the pole. In this case, the angle of elevation is 64 degrees.
Using trigonometry, we can determine that the height of the pole can be found using the equation:
Height = Length of the shadow * tan(angle of elevation)
Given that the length of the shadow is 34 feet and the angle of elevation is 64 degrees, we can plug in these values:
Height = 34 ft * tan(64 degrees)
Calculating this value:
Height ≈ 34 ft * 2.097
Height ≈ 71.31 ft
Now, we have the height of the pole. However, the question asks for the length of the pole. To calculate that, we need to account for the pole's tilt of 12 degrees.
Since the pole is tilted away from the sun, we can imagine a right-angled triangle formed with one leg being the height of the pole and the other leg being the shadow on the ground. The angle between the shadow and the height of the pole is 12 degrees.
Using trigonometry again, we can determine that the length of the pole can be found using the equation:
Length of the pole = Height / cos(angle of tilt)
Given that the angle of tilt is 12 degrees and the height is 71.31 ft, we can plug in the values:
Length of the pole = 71.31 ft / cos(12 degrees)
Calculating this value:
Length of the pole ≈ 71.31 ft / 0.978
Length of the pole ≈ 72.96 ft
Therefore, the length of the pole is approximately 72.96 feet.
To solve this problem, we can use trigonometric ratios such as sine, cosine, and tangent.
First, let's label the given information:
Angle of tilt from the vertical = 12 degrees
Length of the shadow = 34 feet
Angle of elevation from the end of the shadow to the top of the pole = 64 degrees
To find the length of the pole, we need to find the height of the pole above the ground. We can use the tangent ratio to do this.
Step 1: Find the height of the pole above the ground:
Let's call the height of the pole h.
Using the tangent ratio:
tan(64 degrees) = h / 34
h = 34 * tan(64 degrees)
h ≈ 74.70 feet
So, the height of the pole above the ground is approximately 74.70 feet.
Step 2: Find the length of the pole:
To find the length of the pole, we need to include the part of the pole that is below the ground level, which is equal to the length of the shadow.
The total length of the pole is the sum of the height above the ground and the length of the shadow:
Length of the pole = height + length of the shadow
Length of the pole = 74.70 + 34
Length of the pole ≈ 108.70 feet
Therefore, the length of the pole is approximately 108.70 feet.