Question

In a cuboid, area of 3 adjacent faces are 4a^2b^2, 6a^3b^2 and 6a, then the volume of the cuboid is
1. 12a^2b^2
2. 12a^3b^2
3. 6a^3b^2
4. 12a^2b^3

Answer 1

A cuboid has a length 'l', a width 'w' and a height 'h'.

Volume of a cuboid is l*w*h.

The surface area for 3 adjacent faces are: l*w, l*h and w*h.

It is given that the area for 3 adjacent faces are $4a^2b^2,6a^3b^2$ and $6a.$

Assume that $l*w=4a^2b^2\\l=\frac{4a^2b^2}{w}$

Assume that $l*h=6a^3b^2\\l=\frac{6a^3b^2}{h}$.

Therefore, $l=\frac{4a^2b^2}{w}= \frac{6a^3b^2}{h}\\4a^2b^2h=6a^3b^2w\\2h=w*3a\\h=w(\frac{3}{2})a$.

Therefore, we deduct that the final surface area will be $h*w=6a$.

Substitute by $h=w(\frac{3}{2}a)$:

$6a=w^2(\frac{3}{2}a)\\w^2=4\\w=2$.

Now try to get the volume of the cuboid:

$volume=l*h*w\\=\frac{4a^2b^2}{w}*w(\frac{3}{2}a)*w\\=4a^2b^2*\frac{3}{2}a*2\\=12a^3b^2$.

Therefore, the correct answer is 2)$12a^3b^2$

Answer 2

Let length of cuboid is L
width of cuboid is B
and height of cuboid is H

if we assume , a and b both are positive term then, 6a³b² should be area of the face which are enclosed by length and width of cuboid.
so, L × B = 6a³b² ......(1)

similarly, we can assume L × H = 4a²b² ...(2)
and B × H = 6a ......(3)

now, multiply all theses equations
(L × B) × (L × H) × (B × H) = (6a³b²) × (4a²b²) × (6a)

(LBH)² = 144a^6b^4

taking square root both sides,

LBH = 12a³b²

hence, volume of cuboid = 12a³b²
so, option (2) is correct.