Question

A sign standing 6 feet tall is situated 18 feet away from a flag pole. at a certain time of day the sign's shadow is 3 feet long and the flagpole's shadow is 24 feet long. how tall is the flagpole?

Answer 1

Observe attached picture.

On picture we have:
A = height of flagpole = x ft
B = length of flagpole's shadow = 24 ft
C = height of sign = 6 ft
D = length of sign's shadow = 3 ft

When we draw a picture representing this problem we can also add another line marked in red. This way we can see that we have two right-angle triangles. We can see that both have same angle marked with α.

We can apply trigonometry rules to find height of flagpole.

From small triangle containing sign we can find tangens function:
$tan \alpha = \frac{C}{D}$
Similarly we can do for large triangle containing flagpole:
$tan \alpha = \frac{A}{B}$

We see that these two equations have same left sides. This means that their right sides must also be same:
$\frac{C}{D} = \frac{A}{B}$
We can solve for A:
$CB=AD \\ A= \frac{CB}{D} \\ A= \frac{6*24}{3} \\ A=48 ft$

Height of flagpole is 48 feet.