Problem — Volume of a rectangular parallelepiped

Problem

A rectangular parallelepiped of aluminium has edges $a = 20\;\text{cm}$, $b = 30\;\text{cm}$, $c = 20\;\text{cm}$. Calculate the volume in cubic metres and express it in scientific notation.

Solution

Step 1 — Converting the edges to metres. Recalling that $1\;\text{cm} = 10^{-2}\;\text{m}$:

$a = 0{,}20\;\text{m}, \qquad b = 0{,}30\;\text{m}, \qquad c = 0{,}20\;\text{m}$

Step 2 — Volume. The volume of a rectangular parallelepiped is the product of the three edges:

$V = a\,b\,c = 0{,}20\cdot 0{,}30\cdot 0{,}20\;\text{m}^3 = 0{,}012\;\text{m}^3$

Step 3 — Scientific notation. Shifting the decimal point by two places:

$\ev{V = 1{,}2\cdot 10^{-2}\;\text{m}^3}$

Dimensional check. The product of three lengths has dimensions $[L^3]$, consistent with a volume. As a cross-check in cubic centimetres: $V = 20\cdot 30\cdot 20 = 12\,000\;\text{cm}^3 = 12\,000\cdot 10^{-6}\;\text{m}^3 = 1{,}2\cdot 10^{-2}\;\text{m}^3$. ✓

Linked atoms

Argomenti: Physical quantities and measurement Concetti: Scientific notation and orders of magnitude Competenze: Unit conversion