Ratio Question 3 with Solutions

Earlier this week I went over my method for setting up a ratio, and I gave you three ratio problems to try on your own. You can see that post here: Setting Up a Ratio

Today I would like to give a solution to the third of those three ratio problems. You can see solutions for the first and second problems here: Ratio Q1 Ratio Q2

Problem 3: The height of a solid cone is 22 centimeters and the radius of the base is 15 centimeters. A cut parallel to the circular base is made completely through the cone so that one of the two resulting solids is a smaller cone. If the radius of the base of the small cone is 5 centimeters, what is the height of the small cone, in centimeters?

For those students that are getting the hang of this, here’s the quick computation right away:

Quick solution:  (22/15) ⋅ 5 = 22/3.

And here are the details for the rest of us:

Detailed solution. A picture of the problem looks like this:

cone

In the above picture we have the original cone together with a cut forming a smaller cone. We have also drawn two triangles that represent the 2 dimensional cross sections of the 2 cones. Let’s isolate the triangles:

The two triangles formed are similar, and so the ratios of their sides are equal. We identify the 2 key words “height” and “radius.”

 height            22       h
radius             15      5

We now find h by cross multiplying and dividing.

22/15 = h/5
110 = 15h
= 110/15 = 22/3

 

Definition: Two triangles are similar if they have the same angles.

Notes: (1) To show that two triangles are similar we need only show that two angles of one of the triangles are equal to two angles of the other triangle (the third is free because all triangles have 180 degrees).

(2) Similar triangles do not have to be the same size.

(3) Similar triangles have sides that are all in the same proportion.

 

More Problems with Explanations

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