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A student is trying to remember a formula from geometry.
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Assuming that π΄ corresponds to area, π corresponds to volume, and all other variables are lengths.
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What missing dimension in the formula π equals π΄ must the right-hand side of the equation be multiplied by to make the formula dimensionally consistent?
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Considering these two terms, π΄ corresponding to area and π to volume, if we write out their dimensions, the dimensions of π΄ are length squared, while the dimensions of π are length cubed.
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Considering the formula π equals π΄ multiplied by something, weβre asked what dimension needs to fit in to that blank spot.
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If we look at the equation from a dimension perspective, we can see that, on the left-hand side, we have πΏ cubed.
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And currently, on the right-hand side, we have πΏ squared.
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So weβre missing a dimension of πΏ.
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This means we would want to multiply the right-hand side of our equation by a length πΏ in order to make it dimensionally consistent.
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And with this πΏ in place, our equation is.