Maths Olympiad Prep

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, 2012

Algebra Difficulty 1.7 Junior Find the answer Canada

All three scales shown are balanced.

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One possible replacement for the ? is

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A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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 cm3^3, then the volume of this rectangular prism is 2×120=2402\times120=240 cm3^3. The volume of the rectangular prism equals the area of its base 3×43\times 4 times its height, hh. Since the volume is 240, then 3×4×h=2403\times 4 \times h=240 or 12h=24012h=240, so h=24012=20h=\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.

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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.