Question

A pole that is 20 feet tall casts a shadow 48 feet long. At the same time, another pole casts a shadow having a length of 12 feet. What is the height, in feet, of the second pole?

Answer 1

you can setup a proportion to figure out the height of the pole (we'll call it X).

20/24 = X/12

then solve for x:

48X = (20*12)
48X = 240
X = 5 feet tall

Answer 2

The "setup in proportion" is nothing else but the use of similarity of two triangles.

Answer 3

woops, typo:

20/48 = X/12 (in the first line)

Answer 4

The ratio of the first pole to its shadow must be the same as the ratio of the second pole to its shadow. Here we can derive an equation (where h=the height of the second pole):
\[\frac{20}{48}=\frac{h}{12}\]
Simplify the first fraction to get that
\[\frac{5}{12}=\frac{h}{12}\]
Multiply each side by 12 to get that h=5
So the second pole must be 5 feet long.

Answer 5

The ratio of the first pole to its shadow must be the same as the ratio of the second pole to its shadow. -- Why ?

Answer 6

(It's not drawn to scale)
Anyways, the top angles we have to assume are the same, because the sun is shining down on both of them at the same angle. Assuming the poles are perpendicular to the ground, we get 2 right triangles formed by the poles, their shadows, and the invisible line connecting the end of the pole to the end of the shadow. Corresponding ratios in similar triangles must be equal :)