Question

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 and the total surface area is 312. The length of the longest edge is

• A. 12
• B. 16
• C. 18
• D. 20

Answer 1

The correct option is C

18

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

Volume = $\frac{a}{r}$ $\times$ a $\times$ ar

$\Rightarrow$ $a^3$ = 216

$\Rightarrow$ a = 6

Surface area = ( $\frac{a}{r}$ $\times$ a + a $\times$ ar + $\frac{a}{r}$ $\times$ ar) $\times$ 2

= 312 = 2 $a^2$( $\frac{1}{r}$+ r + 1)

r + $\frac{1}{r}$ + 1 = $\frac{312}{2\times 36}$

r + $\frac{1}{r}$ + 1 = $\frac{13}{3}$

$r^2$ + 1 + r = $\frac{13r}{3}$

3$r^2$ + 3 + 3r = 13r

$\Rightarrow$ 3$r^2$ - 10r + 3 = 0

3$r^2$ - 9r - r + 3 = 0

3r(r - 3) - r(r - 3) = 0

(3r - 1)(r - 3) = 0

r = $\frac{1}{3}$ or 3

Taking r =3

The sides are 2, 6, 18

$\Rightarrow$ longest side = 18