Question

The volume of a cuboid is `24sqrt(42)` cm³. Its length is `5sqrt(2)` cm, breadth and height are in the ratio `sqrt(3) : sqrt(7)`. The height of the cuboid is ______.

Answer 1

The volume of a cuboid is `24sqrt(42)` cm³. Its length is `5sqrt(2)` cm, breadth and height are in the ratio `sqrt(3) : sqrt(7)`. The height of the cuboid is `underlinebb((2sqrt(210))/5 cm)`.

Explanation:

Given: Volume `V = 24sqrt(42)` cm³

length `l = 5sqrt(2)` cm

breadth : height = `sqrt(3) : sqrt(7)`

Step wise calculation:

1. Let breadth = `ksqrt(3)` and height = `ksqrt(7) (k > 0)`.

2. Volume = l·b·h

= `(5sqrt(2))(ksqrt(3))(ksqrt(7))` 

= `5kroot(2)(2·3·7)`

= `5kroot(2)42`

3. Set equal to given volume:

`5kroot(2)(42) = 24sqrt(42)`

⇒ 5k² = 24

⇒ `k^2 = 24/5`

⇒ `k = sqrt(24/5)`

4. Height `h = ksqrt(7)` 

= `sqrt(24/5)·sqrt(7)` 

= `sqrt(168/5)`

= `(2sqrt(210))/5` cm