From the top of a hill 700 ft. high, an observer notes that the angles of depression of two successive markerstones on a level ground, which leads straight away from the hill are 5 degree and 15 degree respectively. Find the distance between the two markerstones.
tan 15 = 700/D1
tan 5 = 700/D2
find D1 and D2
subtract
To find the distance between the two markerstones, we can use trigonometric ratios.
Let's assume that the observer is at point A at the top of the hill, and the two markerstones are at points B and C on the level ground.
First, let's draw a diagram to help visualize the situation:
B
/ |
700 / |
/ |
/ |
/15° |
/ |
A /_________| C
x
Given that the angle of depression from point A to point B is 5 degrees, we can use the tangent function to find the distance AB.
Tangent of angle 5 degrees is equal to the opposite side (AB) divided by the adjacent side (700 ft):
tan(5°) = AB / 700
To find AB, we can rearrange the equation:
AB = tan(5°) * 700
Next, we can find the distance AC by using the tangent function with the angle of depression of 15 degrees:
tan(15°) = AC / 700
Rearranging the equation to solve for AC:
AC = tan(15°) * 700
Finally, we can find the distance between the two markerstones BC by subtracting AB from AC:
BC = AC - AB
Now, we can calculate the values:
AB = tan(5°) * 700 ≈ 60.18 ft
AC = tan(15°) * 700 ≈ 191.15 ft
BC = AC - AB ≈ 130.97 ft
Therefore, the distance between the two markerstones is approximately 130.97 ft.