Question

If V is the volumw of cuboid of dimensions a,b,c and S is its surface area, then prove that 1/V = 2/S( 1/a + 1/b + 1/c)

Answer 1

$\textbf{Given:}$

$\text{Surface area of cuboid, S}=2(ab+bc+ca)$

$\text{Volume of cuboid, V}=abc$

$\text{Now,}$

$S=2(ab+bc+ca)$

$\text{This can be written as}$

$S=\displaystyle2(\frac{abc}{c}+\frac{abc}{a}+\frac{abc}{b})$

$\text{Taking abc as common}$

$S=\displaystyle2(abc)(\frac{1}{c}+\frac{1}{a}+\frac{1}{b})$

$S=\displaystyle2(V)(\frac{1}{c}+\frac{1}{a}+\frac{1}{b})$

$\frac{S}{2}=\displaystyle\,V(\frac{1}{c}+\frac{1}{a}+\frac{1}{b})$

$\implies\boxed{\bf\frac{1}{V}=\frac{2}{S}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}$