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

Answer 1

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
$\frac{a}{r}\times a\times ar\left[\because \mathrm{volume}=\mathrm{length}\times \mathrm{breadth}\times \mathrm{height}\right]$
i.e a³ = 216 (given)
i.e a = 6
Since surface area of rectangular block
$$\begin{gathered} =2\left(\frac{a}{r}·a+a.ar+\frac{a}{r}ar\right)=252\left(\mathrm{given}\right) \\ \left(\because \mathrm{surface}\mathrm{area}=2\left(lb+bh+hl\right)\right) \\ \Rightarrow 2\left(\frac{a^2}{r}+a^{2}r+a^2\right)=252 \\ \Rightarrow 2a^2\left(\frac{1}{r}+r+1\right)=252 \\ \Rightarrow 2\times 36\left(\frac{1+r^2+r}{r}\right)=252 \\ \Rightarrow 2\left(1+r^2+r\right)=7r \\ \mathrm{i}.\mathrm{e}2r^2+2+2r=7r \\ \mathrm{i}.\mathrm{e}2r^2-5r+2=0 \end{gathered}$$

$$\begin{gathered} \mathrm{i}.\mathrm{e}\left(2r-1\right)\left(r-2\right)=0 \\ \Rightarrow r=\frac{1}{2},r=2 \\ \therefore \mathrm{for}r=\frac{1}{2},\mathrm{length}=\frac{a}{r}=\frac{6}{{\displaystyle \frac{1}{2}}}=12 \end{gathered}$$
Breadth = a = 6
Height = 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.