Peace to all. After gathering and calculating the measurements by volume (of a box), will the results be in regular inches or will they be in inches^3?
For example, if the dimensions of the item are as follows: L = 4.13in, W = 5.50in, H = 7.50in, will the result of the volume be 170.36in or 170.36in^3
When is inches^3 appropriate compared to inches?....Does the same rule apply to all measurements in this scenario (such as CM, MM, etc)?
On
Indeed the measurement of the volume of the box will be in inches cubed, i.e. $\text{in}^3$.
The same goes for other units: If the side length of the box is in centimeters ($\text{cm}$), then the volume of the box will have units $\text{cm}^3$.
On
If $X$ is the number of cubic units $\space (u)\space$ it is always appropriate to write
$ X\space u^3\space$ as opposed to $X^3 u.\quad$ The distinction is important where, for example, $X>1\implies X^2 mi >Xmi^2.\quad$ Oddly enough, casual language portrays it the other way: "$X\space$ square miles is less than $X\space$ miles squared," where the latter refers to a square that is $X$ miles on a side.
In your example each $1D$ measurement is expressed as a unit to the first power but the final expression shows the unit cubed. $$4,13\space in\space \times\space 5.5\space in\space \times\space 7.5\space in = 170.3625\space in^3$$
On
A volume is always measured in length3. In your case, length is inches.
Outside of physics, people may not be quite so careful, or knowledgable. If you buy a chainsaw, you might be told it has a 1.6" engine. In context, it obviously means cubic inches. You're smart enough to figure that out. The chainsaw salesman may neither appreciate nor understand a correction.
You can sort of treat the units as quantities. The volume is $4.13\text{ in}\times 5.50\text{ in}\times 7.50\text{ in}=(4.13\times 5.50\times 7.50)(\text{in})^3$. Hence, the unit of volume is $\text{inches}^3$. In general, you can apply this strategy of treating units as a variable/quantity in order to find the units for another property/measurement calculated from a formula. It can also be helpful when figuring out unit conversions.