Question

The lengths of three unequal edges of a rectangular solid block are in GP. The volume and total surface area of the block are $216{\text{cm}}^3$ and $252{\text{cm}}^2$, respectively. Find the shortest edge of the block.

• A. $12$ cm
• B. $6$ cm
• C. $18$ cm
• D. $3$ cm

Answer 1

Answer: (D)

$3$ cm

Let the edges of the block be $\frac{a}{r},a,ar$

Therefore, $\frac{a}{r}.a.ar=216$

$\Rightarrow a^3=216$

$\Rightarrow a=6$

And $2[\frac{a}{r}+a.ar+ar.\frac{a}{r}]=252$

$\Rightarrow 2a^2[\frac{1}{r}+r+1]=252$

$\Rightarrow \frac{1}{r}+r+1=\frac{252}{2\times 6\times 6}=\frac{7}{2}$

$\Rightarrow r+\frac{1}{r}=\frac{5}{2}$

$\Rightarrow r=2$

Thus shortest edge is $\frac{a}{r}=\frac{6}{2}=3$ cm.