Question

A flagpole 8.2 meters tall casts a horizontal shadow 4.0 meters long. What time is it if the sun was directly overhead at 12:10 pm and will set at 6:10 pm?

Answer 1

1:54 pm

Explanation:

This solution is proposed supposing that from zenith to horizon the sun (or better, the earth) describes an arc of `90^o` (if there are hills to the horizon, it may not be true). This path takes 6 hours (from 12:10 pm to 06:10 pm) to be completed.

Now the angle (x) between the flagpole and its shadow, after 12:10 pm, is the same angle in which the sun is above the horizon. Since the flagpole is at `90^o` to the ground, the height of the flagpole and its shadow are the legs of a right triangle in which
`tan x = "opposed cathetus"/"adjacent cathetus"`
`tan x = "flagpole's height"/"shadow's horizontal length"`
`tan x=8.2/4=2.05` => `x=64^o`

Knowing that the sun, at the time, was `64^o` above the horizon, we can determine the time (`Delta t`) till the sunset:
`Delta t= (64^o/90^o)*6" hours"=4.26666" hours"`
The fraction of hours we can convert in minutes:
#0.26666" hours"=0.26666*60" minutes"=16 " minutes"#

So the time was 4 hours and 16 minutes before sunset or:
`t=6:10 "pm" - 4:16 = 1:54 "pm"`