A vertical pole is standing on a slope that is inclined at an angle 8 degrees to the horizontal and casts a shadow that is 82 ft. long down the slope. How tall is the pole if the angle of elevation of the sun is 28 degrees?
Draw a diagram. If the pole has height h, and the tip of the shadow is x horizontally away from the pole, then
(82 sin8° + h)/x = tan28°
x/82 = cos8°
Now, eliminate x to get
(82 sin8° + h)cot28° = 82 cos8°
Now just solve for h
Well, let me calculate that for you. But first, let me check my vertical inclinometer... Ah, found it! *rummages around in a clown's pocket* Now let's see... Ah yes, the height of the pole.
Since we know the angle of elevation of the sun is 28 degrees, and the pole's shadow on the slope is 82 ft long, we can use some trigonometric wizardry to find the height of the pole.
The tangent of the angle of elevation is equal to the opposite side (the height of the pole) divided by the adjacent side (the shadow on the slope). So, using some clown-calculations, we get:
tan(28) = height / 82
Now, let's solve for the height:
Height = tan(28) * 82
According to my clown-calculator, the height of the pole is approximately 51.38 feet.
So, the pole stands tall and proud at about 51.38 feet, defying gravity on that slope!
To find the height of the pole, we can use the concept of similar triangles.
Let's call the height of the pole "h."
We have two right triangles: one formed by the pole, its shadow, and a horizontal line, and another formed by the slope, the pole's shadow, and the horizontal line.
Angle of elevation of the sun: 28 degrees
Angle of inclination of the slope: 8 degrees
Since the angle of elevation and the angle of inclination are complementary angles (sum up to 90 degrees), we can find the angle of inclination of the pole with respect to the horizontal by subtracting the angle of elevation from 90 degrees.
Angle of inclination of the pole = 90 - 28 = 62 degrees
Using the tangent function, we can set up the following proportion:
tan(62 degrees) = h / 82 ft
Solving for h, we get:
h = tan(62 degrees) * 82 ft
Using a calculator, we find:
h ≈ 153.24 ft
Therefore, the height of the pole is approximately 153.24 ft.
To find the height of the pole, we can use trigonometry. Let's break down the problem step by step:
Step 1: Identify the given information:
- The angle of elevation of the sun is 28 degrees.
- The slope is inclined at an angle of 8 degrees to the horizontal.
- The length of the shadow cast by the pole down the slope is 82 ft.
Step 2: Draw a diagram:
You can draw a diagram to visualize the problem. Draw a vertical line to represent the pole, a horizontal line to represent the ground, and a line from the top of the pole to the edge of the shadow.
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Step 3: Identify the relevant trigonometric ratios:
In this problem, we can use the tangent (tan) function because we have the opposite and adjacent sides of the right triangle formed by the sun, the pole, and its shadow.
The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
Step 4: Set up the trigonometric equation:
Let's label the triangle sides in the diagram:
- The length of the shadow (opposite side) is 82 ft.
- The height of the pole (adjacent side) is what we need to find.
- The angle of elevation of the sun between the ground and the line from the top of the pole to the edge of the shadow is 28 degrees.
Using the tangent function:
tan(28°) = Opposite/Adjacent
tan(28°) = 82 ft/h ft
Where "h" represents the height of the pole.
Step 5: Solve for the height of the pole:
To solve for "h," we rearrange the equation:
h = (Opposite side) / tan(28°)
h = 82 ft / tan(28°)
Using a calculator, evaluate the tangent of 28 degrees:
h ≈ 82 ft / 0.5317
h ≈ 154.2 ft
Therefore, the height of the pole is approximately 154.2 feet.