Question

Find the volume of a cuboid whose diagonal is `3sqrt(29)"cm"` when its length, breadth and height are in the ratio 2 : 3 : 4.

Answer 1

Given that:
Diagonal of cuboid = `3sqrt(29)"cm"`...............................(1)
Ratio of Length, breadth & height = 2 : 3 : 4
∴ Length (l) = 2x
Breadth (b) = 3x  &
Height (h) = 4x
We know that: 
Diagonal of cuboid 
= `sqrt("l"^2 + "b"^2 + "h"^2)`
= `sqrt((2x)^2 + (3x)^2 + (4x)^2)`
= `sqrt(4x^2 + 9x^2 + 16x^2)`
= `sqrt(29x^2)`
= `xsqrt(29)`
Also,
`xsqrt(29) = 3sqrt(29)`    ...[From (1)]

i.e., x = `(3sqrt(29))/sqrt(29)`
∴ x = 3cm
Thus,
Length = 2 x 3 = 6cm
Breadth = 3 x 3 = 9cm
Height = 4 x 3 = 12cm
∴ Volume of cuboid
= l x b x h
= 6 x 9 x 12
= 54 x 12
= 648cm³.