Question

A woman standing on a hill sees a flagpole that she knows is 65 ft tall. The angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. Find her distance x from the pole.

Answer 1

To find the distance x from the pole, we can use trigonometry to set up equations using the tangent function. By solving these equations for the distances to the bottom and top of the pole, we can find the distance x as the difference between the two distances.

Step-by-step explanation:

To find the distance x from the pole, we can use trigonometry. The woman is standing on a hill and sees the flagpole at an angle of depression of 14° to the bottom and an angle of elevation of 18° to the top. Let's call the distance from the woman's position to the bottom of the pole as d1 and the distance from the woman's position to the top of the pole as d2.

Using the tangent function, we can set up the equations:

tan(14°) = 65 / d1 to find d1

tan(18°) = 65 / d2 to find d2

From these equations, we can solve for d1 and d2. Then, the distance x from the pole is simply the difference between the two distances, x = d2 - d1.

Answer 2

Let the distance between woman and pole be "x".

From diagram,

In ΔOAB,

tan 18 =
`(Perpendicular)/(Base)`

tan 18 =
`(65 -y)/(x)`

∴ x =
`(65 -y)/(tan 18)` ................ (1)

In ΔOBC

tan 14 =
`(Perpendicular)/(Base)`

tan 14 =
`(y)/(x)`

y = x × tan 14 ..............(2)

equating (2) in (1), we get;

x =
`(65)/(tan 18 + tan 14)`

x =
`(65)/(0.574)`

∵ tan 18 + tan 14 = 0.574

∴ x = 113.2 ft

i.e. The her distance from the pole is 113.2 ft.