# The lengths of three unequal edges of a rectangular solid block are in G.P.

Question:

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252 cm2. The length of the longest edge is

(a) 12 cm

(b) 6 cm

(c) 18 cm

(d) 3 cm

Solution:

Since lengths of three unequal sides of a rectangular solid block are in G.P.

Let the length, breadth and height of rectangular solid block be $\frac{a}{r}, a$ and ar respectively.

∴ Volume of rectangular block is

$\frac{a}{r} \times a \times a r \quad[\because$ volume $=$ length $\times$ breadth $\times$ height $]$

i.e a3 = 216    (given)

i.e a = 6

Since surface area of rectangular block

$=2\left(\frac{a}{r} \cdot a+a \cdot a r+\frac{a}{r} a r\right)=252 \quad$ (given)

$(\because$ surface area $=2(l b+b h+h l))$

$\Rightarrow 2\left(\frac{a^{2}}{r}+a^{2} r+a^{2}\right)=252$

$\Rightarrow 2 a^{2}\left(\frac{1}{r}+r+1\right)=252$

$\Rightarrow 2 \times 36\left(\frac{1+r^{2}+r}{r}\right)=252$

$\Rightarrow 2\left(1+r^{2}+r\right)=7 r$

i. e $2 r^{2}+2+2 r=7 r$

i. e $2 r^{2}-5 r+2=0$

i.e $(2 r-1)(r-2)=0$

$\Rightarrow r=\frac{1}{2}, r=2$

$\therefore$ for $r=\frac{1}{2}$, length $=\frac{a}{r}=\frac{6}{\frac{1}{2}}=12$

Breadth = a = 6

Height $=\operatorname{ar}=6 \times \frac{1}{2}=3$ and for $r=2$,

length $=\frac{a}{r}=\frac{6}{2}=3$

breadth = a = 6

right = ar = 12

∴ length of largest edge is 12

Hence, the correct answer is option A.

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