The angle of elevation of the top of a vertical pole from a height 1.54m above a horizontal ground is 40 degree.the foot of the pole is on the same horizontal ground and the point of observation is 20m from the pole.calculate,correct to 3 s.f (a)the height of the pole (b)the angle of depression of the foot of the pole from the point of observation
Your solving are not clear enough u should have illustrated with a diagram
Draw a diagram. It should be clear that
(a) h = 1.54 + 20 cot40°
(b) tanθ = 1.54/20
Answer
(a) Well, well, well! Looks like we've got ourselves a pole situation here. Okay, let's do some calculations. We know the angle of elevation from the point of observation to the top of the pole is 40 degrees, and the height above the ground is 1.54m. Now, to find the height of the pole, we need to use a little bit of trigonometry magic.
We can set up a right triangle here, with the height of the pole being the opposite side, and the distance from the point of observation to the pole being the adjacent side. So, we can use the tangent function:
tan(40 degrees) = height of pole / 20m
By rearranging the equation, we can find the height of the pole:
height of pole = tan(40 degrees) * 20m
Now all we need to do is punch in the numbers and get our answer. Calculating, calculating... drum roll, please!
The height of the pole is approximately 13.784 meters.
(b) Ah, the angle of depression. So, we're looking for the angle between the line of sight from the point of observation to the foot of the pole and the horizontal ground. Since the pole is vertical, the angle of depression will be equal to the angle of elevation, just in the opposite direction.
So, the angle of depression from the point of observation to the foot of the pole is also approximately 40 degrees.
Voila! There you have it, my friend. The height of the pole is approximately 13.784 meters, and the angle of depression from the point of observation to the foot of the pole is approximately 40 degrees.
To solve this problem, we can use basic trigonometric concepts and formulas.
(a) To find the height of the pole, we can use the tangent function. The tangent of the angle of elevation is equal to the opposite side divided by the adjacent side. In this case, the height of the pole is the opposite side and the distance from the point of observation to the pole is the adjacent side.
Let's calculate the height of the pole.
Tangent of the angle of elevation = Height of the pole / Distance from the point of observation to the pole
Tan(40 degrees) = Height of the pole / 20 meters
To find the height of the pole, we rearrange the formula:
Height of the pole = Tan(40 degrees) * 20 meters
Now, let's calculate:
Height of the pole = Tan(40 degrees) * 20 meters
Using a calculator, we find that the height of the pole is approximately 14.984 meters.
Therefore, the height of the pole is 14.984 meters (rounded to 3 decimal places).
(b) To find the angle of depression of the foot of the pole from the point of observation, we can use the inverse tangent function. The inverse tangent of a ratio gives us the angle.
The opposite side is the height of the pole, and the adjacent side is the distance from the point of observation to the pole.
Let's calculate the angle of depression.
Angle of depression = Inverse Tan(Height of the pole / Distance from the point of observation to the pole)
Angle of depression = Inverse Tan(14.984 meters / 20 meters)
Using a calculator, we find that the angle of depression is approximately 38.878 degrees.
Therefore, the angle of depression of the foot of the pole from the point of observation is approximately 38.878 degrees (rounded to 3 decimal places).