Home
Class 12
MATHS
Let A(1), A(2), A(3), A(4) be the areas ...

Let `A_(1), A_(2), A_(3), A_(4)` be the areas of the triangular faces of a tetrahedron, and `h_(1), h_(2), h_(3), h_(4)` be the corresponding altitudes of the tetrahedron. If the volume of tetrahedron is `1/6` cubic units, then find the minimum value of ` (A_(1) +A_(2) + A_(3) + A_(4))(h_(1)+ h_(2)+h_(3)+h_(4))` (in cubic units).

Text Solution

Verified by Experts

The correct Answer is:
`8`
Volume `(V)= (1)/(3) A_(1)h_(1) or h_(1) = (3V)/(A_(1))`
Similarly `h_(2)= (3V)/(A_(2)), h_(3)= (3V)/(A_(3)) and h_(4)= (3V)/(A_(4))`
So `" "(A_(1)+ A_(2)+ A_(3)+A_(4))(h_(1)+h_(2)+ h_(3)+h_(4))`
`" "= (A_(1)+A_(2)+A_(3)+A_(4))((3V)/(A_(1))+ (3V)/(A_2)+ (3V)/(A_3)+ (3V)/(A_4))`
`" "= 3V(A_(1)+ A_(2)+A_(3)+A_(4))((1)/(A_1)+ (1)/(A_2)+ (1)/(A_3)+ (1)/(A_4))`
Now using A.M.-H.M inequality in `A_(1), A_(2), A_(3), A_(4)`, we get
`" "(A_1+A_2+A_3+A_4)/(4) ge (4)/(((1)/(A_1)+ (1)/(A_2)+ (1)/(A_3)+ (1)/(A_4)))`
or `" "(A_(1) + A_(2) + A_(3) + A_(4))((1)/(A_1)+ (1)/(A_2)+ (1)/(A_3)+ (1)/(A_4)) ge 16`
Hence, the minimum value of `(A_(1)+ A_(2)+ A_(3)+A_(4))(h_(1)+h_(2)+h_(3)+h_(4)) = 3V(16)= 48V= 48(1//6)=8`
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE|Exercise JEE Previous Year|26 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Exercise (Matrix)|5 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Question Bank|20 Videos

  • TRIGNOMETRIC RATIOS IDENTITIES AND TRIGNOMETRIC EQUATIONS

    CENGAGE|Exercise Question Bank|34 Videos

Similar Questions

Explore conceptually related problems

Let A _(1), A_(2), A_(3), A_(4) be the areas of the triangular faces of a tetrahedron, and be the corresponding altitudes of the tetrahedron. If volume of tetrahedron is 1//6 cubic units, then find the minimum value of (A_(1) + A_(2) + A+_(3)+A_(4)) (h_(1) + h _(2)h _(3) +h_(4)) (in cubic units).

If A_(1), A_(2), A_(3), A_(4) are four A.M’s between 1/2 and 3, then prove A_(1) + A_(2) + A_(3) + A_(4) = 7.

Let A_(1), A_(2), A_(3),…,A_(n) be the vertices of an n-sided regular polygon such that (1)/(A_(1)A_(2))=(1)/(A_(1)A_(3))+(1)/(A_(1)A_(4)). Find the value of n. Prove or disprove the converse of this result.

If a_(1),a_(2),a_(3),a_(4) are are the roots of the equation 3x^(4)-(1+m)x^(3)+2x+5l=0 and Sigma a_(1)=3,a_(1),a_(2),a_(3),a_(4),=10 then (1,m)=

If a_(1), a_(2), a_(3), a_(4), a_(5) are consecutive terms of an arithmetic progression with common difference 3, then the value of |(a_(3)^(2),a_(2),a_(1)),(a_(4)^(2),a_(3),a_(2)),(a_(5)^(2),a_(4),a_(3))| is

Let A be the set of 4 -digit numbers a_(1)a_(2)a_(3)a_(4) where a_(1)>a_(2)>a_(3)>a_(4), then n(A) is equal to

Let (a_(1),a_(2),a_(3),a_(4),a_(5)) denote a re=arrangement of (3,-5,7,4-9), then a_(1)x^(4)+a_(2)x^(3)+a_(3)x^(2)+a_(4)+a_(5)=0 has