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Practice Try it in Solve: America's Dairyland Harder problems from real math contests

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The volume of a box, or rectangular prism, is its length times its width times its height, expressed using the follow literal equation:

$V=l * w * h$ .

The answer will be in cubic units, be it inches, feet, or a made-up unit. It doesn't really matter what the length, width or height is because multiplication is commutative. You could just turn the box for a different perspective, and relabel the edges differently.

What is the volume of a box with length $10 \ \text{cm},$ width $\frac{1}{2} \ \text{cm}$ and height $3 \ \text{cm}?$

Spot the idea in disguise

In a contest problem, this idea hides inside a situation you have not seen before. Learning to spot it is the real skill. These are real problems from the AMC 8, AMC 12 and AIME, the national math competitions for middle and high school students, each with a worked solution on LIVE by Po-Shen Loh.

Cube geometry problems 31 from AMC 8, AMC 12, AIME
Rectangular prism problems 12 from AMC 8, AMC 12, AIME

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Spot the idea in disguise

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