Question

The angle of depression from the top of Pike’s Peak, at an altitude of 14,000 feet, to the city of Colorado Springs is 8o. The horizontal distance from the city to the peak is 58,000 feet. Find the altitude of Colorado Springs.

Answer 1

`\mbox{\large \boxed{\textsf{5, 849 \;feet}}}`

Explanation:

See figure for information pertaining to this explanation

The figure is self explanatory

The triangle ABC forms a right triangle with the vertical leg of length 14000-h and horizontal length of 58000

The angle opposite side 14000 - h = 8°

By the tangent formula

`\tan 8^\circ = (AB)/(AC) = (14000-h)/(58000)`

Switching sides,


`(14000-h)/(58000) = \tan 8^\circ`

`14,000 - h = 58,000 * \tan 8^\circ\\\\14,000 - h = 8,151.36 \approx 8,151`

h = 14,000 - 8,151 = 5,849 feet

So the altitude of Colorado Springs is roughly 5,849 feet

This is close to the published value of 6,035 feet. The differences are due to estimate errors in horizontal distance, peak height and angle of depression