Question

How would you convert 8.50 $in^3$ to $m^3$?

Answer 1

Hint: Try and recall the standard conversion factors for how many centimetres make a metre and how many metres make an inch. Following this, raise the factors to powers of three to obtain the conversion in cubic units of metres and inches. Finally extrapolate this to the given measure in the question to obtain the appropriate solution.

Formula Used:
$1\;in^3 = 16.387 \times 10^{-6}\;m^3 $

Complete Solution:
Let us begin by first understanding the units presented to us.

Inches and metres are units of length. This means that the quantity that they measure represents how long an object is. When cubed (or raised to the power of 3), these units become a measure of volume. This means that the quantity that $in^3$ and $m^3$ measure represents the amount of space that an object occupies. Thus, if the measurement of the length (or sides) of an object was ‘$y$’ inches, then its volume is expressed as $y\times y\times y = y^3\;in^3$, where we multiplied the length three times.

From the question, we thus deduce that we are given a measure of volume that needs to be converted from $in^3$ to $m^3$. To do so, the only way to go about it is to remember the following empirical relations:
$1\;cm=0.01\;m = 10^{-2}\;m$, and $1\;in = 2.54\;cm$.

On cubing the above expressions, we get:
$(1\;cm)^3=(10^{-2}\;m)^3$, and $(1\;in)^3 = (2.54\;cm)^3$
$\Rightarrow 1\;cm^3 = 10^{-6}\;m^3$ and $1\;in^3 = 16.387\;cm^3$

Now, since $1\;cm^3 = 10^{-6}\;m^3$, then $16.387\;cm^3 = 16.387\times 10^{-6}\;m^3$
$\Rightarrow 1\;in^3 = 16.387\times 10^{-6}\;m^3$

We can now extrapolate this relationship to our question.
$\Rightarrow 8.50\;in^3 = 8.50 \times 16.387\times 10^{-6}\;m^3 = 1.39 \times 10^{-4}\;m^3$

Therefore, $8.50\;in^3 = 1.39 \times 10^{-4}\;m^3$.

Note:
It case the conversion relations are a bit difficult to remember, simply take your regular $15\;cm$ or $30\;cm$ ruler which usually has markings in centimetres on one edge and inches on the opposite edge. Look at the marking corresponding to the 1 inch mark, which is usually at $\approx 2.5\;cm$ and convert that to metres and cube the values to obtain the cubic conversion relation between inches and metres.