Point A and B are on the same horizontal line with the foot of a hill and the angles of depression of these points from the top of hill are 32° and 22° respectively. If the distance between A and B is 80 meters, what is the height of the hill?
Tan32=h/x
Tan22=h/(x+80)
(Tan32)(x)=(tan22)(x+80)
Solve for x and then for h
To find the height of the hill, we need to first understand the concept of angles of depression.
The angle of depression is the angle between the line of sight from an observer's eye to an object and a line horizontal to the observer's eye. In this case, the observer is at the top of the hill, and the objects are points A and B on the horizontal line.
Now, let's solve the problem step by step:
1. Draw a diagram: Draw a straight horizontal line to represent the ground. Label one end of the line as point A and the other end as point B. Draw a vertical line from the middle of the line to represent the hill. Label the top of the hill as point T.
2. Label the angles: From point T, draw lines at an angle of 32° downwards to point A and an angle of 22° downwards to point B.
3. Find the right triangle: Draw a straight line connecting points A and B. This line represents the distance between A and B, which is 80 meters. Now, you have a right triangle with its right angle at point T.
4. Identify the trigonometric ratios: We can use the trigonometric ratios tangent and sine to calculate the height of the hill.
- For angle of depression 32°: Let the height of the hill be h1. We have: tan(32°) = h1 / 80
- For angle of depression 22°: Let the height of the hill be h2. We have: tan(22°) = h2 / 80
5. Solve for the heights: From the equations above, we can find the values of h1 and h2.
- h1 = 80 * tan(32°)
- h2 = 80 * tan(22°)
6. Find the height of the hill: The height of the hill is the difference between h1 and h2:
- Height of the hill = h1 - h2
By substituting the values of h1 and h2 into the equation and performing the calculation, you can find the height of the hill.