Question

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cubic units and the total surface area is 312 square units. 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 units, 6 units, 18 units

$\Rightarrow$ longest side = 18 units