7th grade Math question - Test review - Please help I'm a bit stuck - Question in desc

Question

A telephone pole casts a shadow that is 20 feet long. At the same time, a 5-foot-tall woman standing on a 1.5-foot-tall podium next to the telephone casts a shadow that is 15 feet long. How tall is the telephone pole?

Answer 1

This is a triangle question. When you stand on the ground next to a taller friend and the sun makes a shadow for each of you, the ratio of your shadow lengths is the same as the ratio of your heights. What does that mean? How does it help here?

If your shadow is 6 feet long, and your friend's shadow is 8 feet long, the ratio between these lengths is 6/8 (6 divided by 8, or 3/4). This is very useful, because now you know the ratio between your height and your friend's height; it's the same, 3/4. A ratio is very useful, because when you know the ratio, and you know your height, you can easily find your friend's height. You are 3/4 as tall as your friend! If you're only 3 feet tall, your friend is 4 feet tall (3 feet times 4/3). If you are 4 feet tall, your friend is 5 feet 4 inches (that's 4 feet times 4/3, or 5 feet + 1/3 foot, and 1/3 of 12" is 4").

So now you know everything needed to figure this out. Instead of You in this problem, we have a 6 1/2 foot tall object (a woman on a podium, 5 + 1 1/2 feet) casting a 15 foot shadow. Instead of your taller friend, we have a telephone pole casting a 20 foot long shadow. So you have one height, and the two shadow lengths, so you calculate the ratio of shadow lengths (15/20 = 3/4) and just like in the first case, you flip it upside down and multiply times the height of the smaller object: 4/3 times 6.5 feet -- I'll let you calculate that amount to get the final answer you need for the height of the telephone pole. It's actually quite a bit short for a telephone pole!

The one part I didn't explain up there is how to know whether to multiply by the ratio or divide by the ratio to get the answer. This confuses lots of people. To figure out for a case like this, just look at the problem and it needs to make sense. You know that your friend casts a longer shadow than you do, so your friend is taller than you. You need to multiply your height by some ratio, so is it the ratio we calculated (3/4) or the inverse ratio (flipped upside down, 4/3)? It's the bigger one, the one that is Greater Than 1, so that when you multiply times the smaller height, you'll get a bigger height. So for the problem, you don't multiply the woman+podium height by 3/4, you multiply by 4/3 to get a taller height.

That's the common sense way to figure out how to do this. ALWAYS try to make sense of your answer this way... Is my answer for the telephone pole sensible? It the pole higher than the woman on the podium? It should be because it casts a longer shadow.

But you should also know how to do this with UNITS. Then you can double check your answer against the common sense approach above. The ratio of units for the shadows needs to coordinate the ratio of units for the heights, so that you can cancel units from the top and bottom of the equation. For you and your friend, this looks like this:

Your height Your shadow

------------------- = ---------------------------

Friend height Friend's shadow

or you can flip them both upside down and they're still equal.

To solve this for your friend's height, we can "cross multiply": the bottom of one side times the top of the other side = top of first side times bottom of other side:

Your height X Friend's shadow = Friend's height X Your shadow

Then to solve for your Friend's height, you get rid of the parts on the right side you don't want (if you get rid of "Your shadow" on the right side, you'll be left with "Friend's height" over there). So you DIVIDE both sides by that shadow:

Your height X Friend's shadow = Friend's height X Your shadow

----------------------------------- -------------------------------------

Your shadow Your shadow

Then what? On the right side we see "Your shadow" on both top and bottom, which means the right side is just your Friend's height times 1 (your shadow cancels out top and bottom):

Your height X Friend's shadow = Friend's height

-----------------------------------

Your shadow

And that's just what we did up above, multiply your height times the ratio of your friend's shadow to your shadow:

4 feet X 4/3 = Friend's height

Hope that explains how to think about questions like this one.