Question Video: Performing Dimensional Analysis on a Cubic Unit Chemistry

If 1 dm is equal to 10 cm, which of the following is 1 dm³ equal to? [A] 10 cm³ [B] 1,000 cm³ [C] 1,000 cm [D] 100 cm³ [E] 1 cm³

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Video Transcript

If one decimeter is equal to 10 centimeters, which of the following is one decimeter cubed equal to? (A) 10 centimeters cubed, (B) 1,000 centimeters cubed, (C) 1,000 centimeters, (D) 100 centimeters cubed, (E) one centimeter cubed.

In order to solve this problem, we will need to use dimensional analysis. Dimensional analysis is a problem-solving method for converting between two different units. In this method, the original value is multiplied by a conversion factor or several conversion factors if necessary. A conversion factor is a relationship of two equal quantities that have different units. The question provides us with a conversion factor: one decimeter is equal to 10 centimeters.

To perform dimensional analysis, this conversion factor will need to be written as a fraction. We can write this conversion factor as two different fractions, one decimeter per 10 centimeters or 10 centimeters per one decimeter. The fraction that we will choose to solve the problem depends on the original value. The original value, the value that we are trying to convert, is one decimeter cubed. We need to multiply our original value by a conversion factor or factors such that the unit decimeters cubed will cancel. When multiplying or dividing, units will cancel when they appear in both the numerator and denominator.

In the original value, the unit decimeter cubed appears in the numerator. This means that we should choose the conversion factor where the unit decimeters appears in the denominator. We can then proceed to multiply by the correct conversion factor. However, multiplying by the conversion factor only one time will not allow the unit decimeters cubed to be completely canceled.

A decimeter cubed is a volume unit that can be expanded to one decimeter times one decimeter times one decimeter. Here we can see that our unit decimeters in the denominator can only cancel one instance of the unit decimeters in the numerator. This means that we will need to multiply by the conversion factor two more times so that all three instances of the unit decimeter in the numerator will be canceled.

We can then perform the calculation. This gives us the answer 1,000 centimeters times centimeters times centimeters. A unit that is multiplied by itself can be written as a single unit with a power. The power indicates the number of times the unit is multiplied by itself. The unit centimeters appears three times. Thus, the unit can be rewritten as centimeters cubed. By performing dimensional analysis, we have determined that one decimeter cubed is equal to answer choice (B), 1,000 centimeters cubed.