A woman standing on a hill sees a flagpole that she knows is 55 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.
To solve this problem, we can use trigonometry and the properties of right triangles.
Let's start by drawing a diagram to visualize the situation. We have a woman standing on a hill, looking at a flagpole. The distance from the woman to the base of the flagpole is x, and the height of the flagpole is 55 ft.
Now, let's label the angles mentioned in the problem. The angle of depression from the woman to the bottom of the pole is 14°, and the angle of elevation from the woman to the top of the pole is 18°.
We can divide the diagram into two right triangles. Triangle ABC will represent the triangle formed by the woman, the base of the flagpole (B), and the top of the flagpole (C). Triangle ABD will represent the triangle formed by the woman, the base of the flagpole (B), and a point directly below the top of the flagpole (D).
Now, let's focus on triangle ABD. The angle at D is a right angle, and we know the height of the flagpole (55 ft). We want to find the distance x from the woman to the flagpole (BD).
In triangle ABD, we can use the tangent function:
tangent(angle) = opposite / adjacent
The opposite side is the height of the flagpole (55 ft), and the adjacent side is the distance x. The angle is the angle of depression (14°). Plugging in these values, we have:
tan(14°) = 55 / x
Now, let's solve for x:
x = 55 / tan(14°)
Using a calculator, we can find that:
x ≈ 224.07 ft
Therefore, the woman is approximately 224.07 ft away from the flagpole.
To find the woman's distance x from the pole, we can use trigonometry and the concept of similar triangles.
Let's start by drawing a diagram to visualize the situation. Sketch a right triangle with the flagpole as the vertical side, the ground as the horizontal side, and the woman's line of sight as the hypotenuse. Label the height of the flagpole as 55 ft.
Next, let's label the angles mentioned in the problem. 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°. Label these angles accordingly in the triangle.
Now, let's focus on the angle of depression. This angle helps us create a smaller right triangle within the main triangle. The hypotenuse of this smaller triangle is the line of sight of the woman, the vertical side is the height of the pole (55 ft), and the angle of depression is 14°.
Using trigonometry, we can find the length of the adjacent side of this smaller triangle. The adjacent side represents the horizontal distance from the woman to the bottom of the pole. We can use the tangent function since we have the angle and the length of the opposite side (55 ft).
tan(14°) = opposite / adjacent
tan(14°) = 55 ft / adjacent
Now, solve for the adjacent side:
adjacent = 55 ft / tan(14°)
Using a calculator, calculate the value of tan(14°) and then divide 55 ft by that result.
Once you have the value of the adjacent side (the horizontal distance from the woman to the bottom of the pole), you can use this to find the total distance x from the pole. The total distance x is the sum of the horizontal distance from the woman to the bottom of the pole and the horizontal distance from the woman to the top of the pole.
Since the angles of elevation and depression are equal, the angles formed by the horizontal line from the woman to the bottom of the pole and the horizontal line from the woman to the top of the pole are also equal. This creates another right triangle.
Now, let's focus on the angle of elevation. This angle helps us create a second smaller right triangle within the main triangle. The hypotenuse of this smaller triangle is the line of sight of the woman, the vertical side is the height of the pole (55 ft), and the angle of elevation is 18°.
Using trigonometry, we can find the length of the adjacent side of this second smaller triangle. The adjacent side represents the horizontal distance from the woman to the top of the pole. We can use the tangent function since we have the angle and the length of the opposite side (55 ft).
tan(18°) = opposite / adjacent
tan(18°) = 55 ft / adjacent
Now, solve for the adjacent side:
adjacent = 55 ft / tan(18°)
Using a calculator, calculate the value of tan(18°) and then divide 55 ft by that result.
Once you have the value of the adjacent side (the horizontal distance from the woman to the top of the pole), you can add it to the horizontal distance from the woman to the bottom of the pole to find the total distance x from the pole.
Draw a diagram, and you will see that
x cot14° + x cot18° = 55
x = 55/(cot14° + cot18°)