A vertical flagpole is mounted on a hill that makes a 10° angle with the horizontal. If the sun, shining at an angle of elevation of 70°, makes a 13 foot shadow down the hill, how tall is the flagpole

Question

A vertical flagpole is mounted on a hill that makes a 10° angle with the horizontal. If the sun, shining at an angle of elevation of 70°, makes a 13 foot shadow down the hill, how tall is the flagpole?

Answer 1

The best way to solve this problem is to draw a figure where the hill makes a 10° angle with the ground and the sun has an elevation angle of 70° which can be measured at the top of the flagpole since there is a 13 ft projection down onto the hill. So at the tip of the flagpole the angle the sun's elevation makes with the flagpole is 20°. Since the hill makes a 10° angle with the horizontal and that the flagpole makes a 90° angle with the horizontal the triangle formed by the hill and the flagpole has an 80° angle that gives a 100° angle that the hill makes with the flagpole. So the triangle formed by the flagpole, sun's elevation, and the hill contains 100° + 20° +x where x = 60°. The law of sines can be used to find the height of the flagpole. So sin20°/13 = sin60°/h and solving for h the height of the flagpole is 32.92 ft or approximately 33 ft.

Answer 2

tan70=length of shadow/ height of pole

tan70=13/ height of pole

height of pole=10.6