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complete question:
An observer at the top of a 462-ft cliff measures the angle of depression from the top of the cliff to a point on the ground to be 5°. What is the distance from the base of the cliff to the point on the ground? Round to the nearest foot
Answer:
a ≈ 5281 ft
Explanation:
The observer at the top of a 462 ft cliff measures the angle of depression from the top of the cliff to a point on the ground to be 5°.
The angle of depression form the top of the cliff = 5°
The 5° is outside the triangle formed . To find the angle in the triangle we have to subtract 5° from 90°. 90° - 5° = 85° Note sum of an angle on a right angle is 90°.
using SOHCAHTOA principle we can solve for the distance from the base of the cliff to the point on the ground(a)
tan 85° = opposite / adjacent
tan 85° = a / 462
cross multiply
462 × tan 85° = a
a = 11.4300523 × 462
a = 5280.66 ft
a ≈ 5281 ft
The distance from the base of the cliff to the point on the ground is 5,280.7 ft.
The given parameters;
- height of the cliff, h = 462 ft
- angle of depression, = 5⁰
The distance from the base of the cliff to the point on the ground is calculated as follows;
- distance from the base of the cliff to the point on the ground forms the height of the triangles of the right triangle,
- height of the cliff is the base
- angle of the base with the hypotenuse side = 90⁰ - 5⁰ = 85⁰
[tex]tan (85) = \frac{h}{462} \\\\h = 462 \times tan(85)\\\\h = 5,280.7 \ ft[/tex]
Thus, the distance from the base of the cliff to the point on the ground is 5,280.7 ft.
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