two points A and B are 562 meter apart on a level stretch of road leading to a hill . the angle of elevation from A is 26degree 30minutes, and angle of elevation from B is 36degee 40minutes. how high is the hill ?
in your sketch mark the top of the mountain as P
and its base as Q
Label AB = 562 and mark you angles
In triangle ABP
angle A = 22° 30'
angle B = 180° - 30° 40 = 149° 20'
angle C = 180 - A-B = 8° 10'
by the Sine Law, you can find BP , which is the hypotenuse of triangle PBQ
it is then easy to find PQ
Note, I left the angles in DMS format, and used the DMS key on my calculator
otherwise change your angles to decimals
To find the height of the hill, you can use trigonometry. Let's break down the steps:
1. Draw a diagram: Draw a triangle with points A, B, and the top of the hill (let's call it C). Label the distance between A and B as 562 meters.
2. Identify the angles: From point A, the angle of elevation to the top of the hill (angle CAB) is 26 degrees 30 minutes. From point B, the angle of elevation to the top of the hill (angle CBA) is 36 degrees 40 minutes.
3. Calculate the distances: Since the base of the hill is the line segment AB, each angle will have its own corresponding distance to the hill.
- From point A to the top of the hill, AC, is the height we want to find.
- From point B to the top of the hill, BC, is also unknown.
4. Use the tangent function: The tangent function relates the ratio of the opposite side to the adjacent side of a right triangle.
- For angle CAB (26 degrees 30 minutes), we can use the tangent function to find AC:
tangent (26.5°) = AC / AB
- Similarly, for angle CBA (36 degrees 40 minutes), we can use the tangent function to find BC:
tangent (36.67°) = BC / AB
5. Solve the equations: Substitute the given values into the tangent equations and solve for AC and BC.
- AC = AB × tangent (26.5°)
- BC = AB × tangent (36.67°)
6. Calculate the height: To find the height of the hill, we need to subtract the difference between AC and BC.
- Height of the hill = AC - BC
By following these steps and plugging in the numbers, you should be able to find the height of the hill.