Question

A flagpole casts a shadow 12 feet long at the same time that a vertical sign 8 feet tall casts a shadow 3 feet long. how high is the flagpole ?

Answer 1

`The flagpo` `l` `e` `color(white)()is 32  feet  high.`

Explanation:

This is a Ratio and Proportion question.

The relationship between the height of the sign and its shadow is the same as the relationship between the flagpole and its shadow.

That relationship is 8:3.
That means that for every 8 feet tall an object is, its shadow is 3 feet long.

If some object is 16 feet tall, its shadow is 6 feet long.

`(object)/(shadow) = (16)/(6) = (8)/(3)`

If some object is 4 feet tall, its shadow is 1.5 feet long.

`(object)/(shadow) = (4)/(1.5) = (8)/(3)`

That works backwards too.
If the shadow is 9 feet long, the object is 24 feet tall.

`(object)/(shadow) = (24)/(9) = (8)/(3)`

Keep the shadow in the denominator even it the problem names it first.

~ ~ ~ ~ ~ ~ ~ ~ ~

In the problem, the ratio of object to shadow is 8:3 in both cases, even though only one object is exactly 8 feet tall.

`(object)/(shadow) = (8)/(3) = (x)/(12)`

That means you can solve for `x` to find the height of the flagpole.

You can solve the pair of equal fractions by cross multiplying and then solving for `x`, already defined as "the height of the flagpole."

1) Cross multiply

`3x = 8*12`

2) Divide both sides by 3 to isolate `x`, already defined as "the height of the flagpole.

To weasel out of having to multiply 8`*`12, make the numbers small by first dividing by 3.

After you have divided both sides by 3, you will have this:

`x = 8*4`

`x = 32` `larr` answer for "the height of the flagpole"

Answer:

`The  flagpo` `l` `e` `is  32  feet  high.`