Video: Using Dimensional Analysis to Determine the Consistency of Given Expressions

A student is trying to remember a formula from geometry. Assuming that 𝐴 corresponds to area, 𝑉 corresponds to volume, and all other variables are lengths, what missing dimension in the formula 𝑉 = 𝐴 must the right-hand side of the equation be multiplied by to make the formula dimensionally consistent?

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Video Transcript

A student is trying to remember a formula from geometry. Assuming that 𝐴 corresponds to area, 𝑉 corresponds to volume, and all other variables are lengths. What missing dimension in the formula 𝑉 equals 𝐴 must the right-hand side of the equation be multiplied by to make the formula dimensionally consistent?

Considering these two terms, 𝐴 corresponding to area and 𝑉 to volume, if we write out their dimensions, the dimensions of 𝐴 are length squared, while the dimensions of 𝑉 are length cubed. Considering the formula 𝑉 equals 𝐴 multiplied by something, we’re asked what dimension needs to fit in to that blank spot. If we look at the equation from a dimension perspective, we can see that, on the left-hand side, we have 𝐿 cubed. And currently, on the right-hand side, we have 𝐿 squared. So we’re missing a dimension of 𝐿.

This means we would want to multiply the right-hand side of our equation by a length 𝐿 in order to make it dimensionally consistent. And with this 𝐿 in place, our equation is.

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