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DABC be a tetrahedron such that AD is pe...

DABC be a tetrahedron such that AD is perpendicular to the base ABC and `angle ABC=30^(@)`. The volume of tetrahedron is 18. if value of `AB+BC+AD` is minimum, then the length of AC is

A

`6sqrt(2-sqrt(3))`

B

`3(sqrt(6)-sqrt(2))`

C

`6sqrt(2+sqrt(3))`

D

`3(sqrt(6)+sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
Volume `=1/3AD(1/2AB.BCsin30^(@))`
`rArr 18-1/12(AD.AB.BC)`
`rArr AD.AB.BC=216`
Now, `AB+BC + ADge3 (AD.AB.BC)^(1//3)`

`rArr AB+BC+AD ge18`
Minimum value occurs when AB=BC=AD=6
Hence, `AC = sqrt(AB^(2)+BC^(2)-2AB.BC. cos30^(@))`
`=6sqrt(2-sqrt(3))`
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