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. (Round your answer to one decimal place.)
Draw the woman on the hill with her line of sight touching the pole somewhere in the middle.
label her line of sight x, the top part of the pole h and the bottom part of the pole 65-h
You now have 2 right-angled triangles
for the top one:
tan18° = h/x
h = xtan18
for the bottom part:
tan14 = (65-h)/x
xtan14 = 65-h
h = 65 - xtan14
then xtan18 = 65 - xtan14
xtan18 - xtan14 = 65
x(tan18-tan14) = 65
x = 65/(tan18+tan14) = 113.2
The - symbols in the first and second lines before the last should be +
To find the distance (x) from the pole, we can use trigonometry.
Let's consider the triangle formed by the woman, the top of the flagpole, and the bottom of the flagpole.
We have the following information:
- The height of the flagpole (opposite side): 65 ft.
- The angle of elevation to the top of the pole (angle A): 18°.
- The angle of depression to the bottom of the pole (angle B): 14°.
To find the distance (x), we'll use tangent since we have the opposite and adjacent sides.
Using the tangent function for angle A (18°):
tan(A) = opposite/adjacent
tan(18°) = 65/x
Rearranging the equation, we get:
x = 65 / tan(18°)
Calculating x:
x ≈ 65 / (0.32492)
x ≈ 200.05 ft
Therefore, the woman is approximately 200.05 ft away from the flagpole.
To find the woman's distance from the pole, we can use the tangent function.
First, let's consider the angle of depression. The angle of depression is the angle between the horizontal line from the woman's eye level to the bottom of the pole and the line of sight from her eye to the bottom of the pole. We can use the tangent function to relate this angle to the distance x.
tan(angle of depression) = opposite / adjacent
In this case, the opposite side is the height of the pole (65 ft) and the adjacent side is the distance x.
So, we can write the equation as:
tan(14°) = 65 / x
Next, let's consider the angle of elevation. The angle of elevation is the angle between the horizontal line from the woman's eye level to the top of the pole and the line of sight from her eye to the top of the pole. We can use the tangent function to relate this angle to the distance x.
tan(angle of elevation) = opposite / adjacent
In this case, the opposite side is the height of the pole (65 ft + the woman's eye level) and the adjacent side is the distance x.
So, we can write the equation as:
tan(18°) = (65 + eye level) / x
Now, we have a system of two equations:
1) tan(14°) = 65 / x
2) tan(18°) = (65 + eye level) / x
We can solve this system of equations to find the value of x. Let's do that:
First, rearrange the equations to solve for x:
1) x = 65 / tan(14°)
2) x = (65 + eye level) / tan(18°)
Since we want to find the value of x, set the right-hand sides of the equations equal to each other:
65 / tan(14°) = (65 + eye level) / tan(18°)
Now, we can solve this equation for the given values of the angles of depression and elevation.
Plug in the given values:
65 / tan(14°) = (65 + eye level) / tan(18°)
Now, solve for eye level:
65 / tan(14°) * tan(18°) = 65 + eye level
Multiply both sides by tan(18°):
65 * tan(18°) / tan(14°) = 65 + eye level
Solve for eye level:
eye level = 65 * tan(18°) / tan(14°) - 65
Now, substitute this value back into either of the two original equations to find x:
x = 65 / tan(14°)
Evaluate this expression to find x. Round your answer to one decimal place.