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, 2012

Geometry Difficulty 1.6 Junior Find the answer Canada

A triangular prism has a volume of 120 cm $^3$ . Two edges of the triangular faces measure 3 cm and 4 cm as shown.

The height of the prism, in cm, is

Pick one

Solutions — 2

Solution 1

The triangular prism given can be created by slicing the 3 cm by 4 cm base of a rectangular prism with equal height across its diagonal.
That is, the volume of the triangular prism in question is one half of the volume of the rectangular prism shown.

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Since the volume of the triangular prism is 120 cm $^3$ , then the volume of this rectangular prism is $2\times120=240$ cm $^3$ .

The volume of the rectangular prism equals the area of its base $3\times 4$ times its height, $h$ .

Since the volume is 240, then $3\times 4 \times h=240$ or $12h=240$ , so $h=\frac{240}{12}=20$ .
Since the height of this rectangular prism is equal to the height of the triangular prism in question, then the required height is 20 cm.

Solution 2

In turn, we may use each of the three known scales to find a way to balance a circle, a diamond and a triangle.

Since many answers are possible, we must then check our solution to see if it exists among the five answers given.

First consider the scale at the top right.

A diamond and a circle are balanced by a triangle.

If we were to add a triangle to both sides of this scale, then it would remain balanced and the right side of this scale would contain what we are trying to balance, a circle, a diamond and a triangle.

That is, a circle, a diamond and a triangle are balanced by two triangles.

However, two triangles is not one of the five answers given.

Next, consider the scale at the top left.

A triangle and a circle are balanced by a square.

If we were to add a diamond to both sides of this scale, then it would remain balanced and the left side of this scale would contain what we are trying to balance, a circle, a diamond and a triangle.

That is, a circle, a diamond and a triangle are balanced by a square and a diamond.

This answer is given as one of the five answers.
In the context of a multiple choice contest, we do not expect that students will verify that the other four answers do not balance a circle, a diamond and a triangle. However, it is worth noting that it can be shown that they do not.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.