I'm envisioning a right triangle, with the bird feeder as the long leg, the shadow as the short leg, and the distance between the top of the bird feeder and the top of the shadow as the hypotenuse.
bird feeder's height = x
shadow's height = half of the bird feeder's height = x/2
hypotenuse = 19 ft
Pythagorean theorem:
a² + b² = c²
a² = c² - b²
x² = 19² - (x/2)²
x² = 361 - x²/4
x² + x²/4 = 361
4(x² + x²/4) = 4(361)
4x² + x² = 1444
5x² = 1444
x² = 1444/5
x² = 288.89
x = √288.89
x = 16.99 or 17 feet
x/2 = 17/2 = 8.5 feet
The height of the bird feeder is 17 feet and the height of the shadow is 8.5 feet.
