from a point A on the side of straight road, the angle of elevation of the top of the electric pole is 15°20'. from a point B on the same side of the road, the angle of elevation of the top of the pole is 10°52'. If the distance of A and B is 50 meters, what is the height of the pole?
review the tangent function. Draw a diagram. You can see that if the pole has height h, then
h cot15°20' + h cot10°52' = 50
That is if A and B are on opposite sides of the pole.
If they are on the same side, then B is farther away than A, and
h cot10°52' - h cot15°20' = 50
So, make up your mind.
To find the height of the pole, we can use the trigonometric concept of tangent.
Let's assume point A is the closer point to the pole and point B is the farther point.
Step 1: Convert the angles from degrees and minutes to decimal degrees:
Angle at point A = 15 degrees + (20/60) degrees = 15.3333 degrees
Angle at point B = 10 degrees + (52/60) degrees = 10.8667 degrees
Step 2: Determine the height of the pole using the tangent function:
Let the height of the pole be h meters.
tan(15.3333 degrees) = h / x1 (where x1 is the distance from point A to the pole)
tan(10.8667 degrees) = h / x2 (where x2 is the distance from point B to the pole)
Step 3: Calculate x1 and x2:
x1 + x2 = 50 meters (Given)
x2 = 50 - x1
Step 4: Substitute x2 in the second equation:
tan(10.8667 degrees) = h / (50 - x1)
Step 5: Rearrange the equation to solve for h:
h = (50 - x1) * tan(10.8667 degrees)
Step 6: Substitute x1 and calculate h:
h = (50 - x1) * tan(10.8667 degrees)
h = (50 - x1) * tan(10.8667 degrees)
h = (50 - 50*tan(15.3333 degrees) / tan(10.8667 degrees)
Calculating h using a calculator gives us:
h ≈ 19.72 meters
Therefore, the height of the pole is approximately 19.72 meters.
To find the height of the pole, we can use trigonometry and the given angles of elevation.
Let's start by drawing a diagram to better understand the problem:
A B
*-----------Pole-----------*
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Road Road
In the diagram, A and B represent the points where the observers are located, and the Pole represents the electric pole. The angle of elevation from A to the top of the pole is 15°20', and the angle of elevation from B to the top of the pole is 10°52'. The distance between A and B is 50 meters.
Now, let's use trigonometry to find the height of the pole.
We can start by calculating the height of the pole from point A. Using the tangent function, we have:
tan(15°20') = height of the pole / distance from A to the pole
To find the value of tan(15°20'), we can convert the angle to decimal form:
15°20' = 15 + (20/60) = 15.3333°
tan(15.3333°) = 0.276
height of the pole / distance from A to the pole = 0.276
Since we know that the distance from A to the pole is 50 meters, we can rearrange the equation to solve for the height of the pole:
height of the pole = tan(15.3333°) * 50
Calculating this expression, we find that the height of the pole from point A is approximately 13.8 meters.
Now, we can repeat the process to find the height of the pole from point B. Using the tangent function again, we have:
tan(10°52') = height of the pole / distance from B to the pole
Converting the angle to decimal form:
10°52' = 10 + (52/60) = 10.867°
tan(10.867°) = 0.191
height of the pole / distance from B to the pole = 0.191
Since the distance from B to the pole is also 50 meters, we can rearrange the equation to solve for the height of the pole:
height of the pole = tan(10.867°) * 50
Calculating this expression, we find that the height of the pole from point B is approximately 9.6 meters.
Therefore, the height of the pole is approximately 13.8 meters when measured from point A and 9.6 meters when measured from point B.