Question

A solid cube is cut into two cuboids exactly at middle as shown in fig.6.31. Find the ratio of the total surface area of the given cube and that of the cuboid.

Answer 1

Let the side of the cube be l cm.
∴ Total surface area of the cube = 6l² cm²
If the cube is cut from the middle, then the breadth of the cuboid is $\frac{l}{2}$ cm.
Also,
Length of the cuboid = Height of the cuboid = Side of the cube = l cm
∴ Total surface area of the cuboid = 2(lb + bh + hl)
= $2\left(l\times \frac{l}{2}+\frac{l}{2}\times l+l\times l\right)$
= 4l² cm²

Now,
$\frac{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cube}}{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cuboid}}=\frac{6l^2{\mathrm{cm}}^2}{4l^2{\mathrm{cm}}^2}=\frac{3}{2}$

Hence, the ratio of the total surface area of the given cube to that of the cuboid is 3:2.