a pole leans away from the sun at an angle of 7 degrees to the vertical. when the elivation of the sun is 48 degrees the pole cast a shadow of 41ft long on level ground. how long is the pole?
Draw a big diagram
You end up with a triangle with:
the horizontal side = 47 ft
the side representing the leaning pole ... call it a
and the angle opposite the pole side = 51°
The pole makes an angle of 7° with the vertical ... so the angle between to pole side and the 47ft side = 90 - 7 = 83°
So that make the angle opposite the 47 ft side = 180 - (51 + 83) = 46°
Now you can use the sine rule to find the length of the pole:
a/sin51° = 47/sin46
a = 47 * sin51 / sin46
a = 50.8 ft
so the length of the pole is 50.8 ft
a is vertically under pole, base to tip
41 - a is shadow tip from under pole tip
height of pole = y cos 7 = .993 y
tan 48 = .993 y/(41-a)
tan 7 = a/y so a = y tan 7
then
tan 48 = .993 y/(41 -y tan 7) solve for y
To find the length of the pole, we can use trigonometry. Let's break down the problem step-by-step:
Step 1: Draw a diagram
Draw a vertical line to represent the pole, and draw a horizontal line to represent the shadow on the ground. The angle between the vertical pole and the ground line should be 7 degrees. Label the height of the pole as "h" and the length of the shadow as 41 ft.
Step 2: Identify the known angles and lengths
We know that the angle of elevation of the sun is 48 degrees, and the angle between the pole and the ground is 7 degrees. The length of the shadow on the ground is 41 ft.
Step 3: Use trigonometry to find the length of the pole
We can use the tangent function to find the length of the pole. The equation is: tan(angle) = opposite/adjacent.
In this case, the angle is 48 degrees, the opposite side is the height of the pole (h), and the adjacent side is the length of the shadow (41 ft).
So, tan(48) = h/41.
Step 4: Solve for the height of the pole
Rearrange the equation to solve for h:
h = tan(48) * 41.
Step 5: Calculate the height of the pole
Using a calculator, find the value of tan(48) and multiply it by 41.
h ≈ 41 * 1.1106 ≈ 45.55 ft.
Therefore, the height of the pole is approximately 45.55 feet.
To find the length of the pole, you can use trigonometry and the concept of similar triangles. Follow these steps:
1. Draw a diagram: Draw a vertical line to represent the pole and a horizontal line to represent the shadow on level ground. Label the angle between the pole and the vertical line as 7 degrees. Also, label the angle between the shadow and the horizontal line as 48 degrees.
2. Recognize the similar triangles: As the sun's rays are parallel, the pole's shadow and the pole itself form similar triangles.
3. Set up a proportion: Let x be the length of the pole. The proportion can be set up as:
(Length of pole) / (Length of shadow) = (Height of pole) / (Length of shadow)
x / 41ft = tan(7 degrees) / tan(48 degrees)
Here, we use the trigonometric ratios of tangent since we have the angle values.
4. Solve the proportion: Rearrange the equation to solve for x by multiplying both sides by 41ft:
x = 41ft * (tan(7 degrees) / tan(48 degrees))
5. Calculate the value: Use a scientific calculator to find the value of x. Round the answer to an appropriate number of significant figures.
By following these steps and performing the calculations, you will be able to determine the length of the pole.