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Most Important Selected Qs for JEE AdvancedMathematicsVector Algebra

Let A₁, A₂, A₃, A₄ be the areas of the triangular faces of a tetrahedron, and h₁, h₂, h₃, h₄ be the corresponding altitudes of the tetrahedron. If volume of tetrahedron is 5 cubic units, then find the minimum value of (A₁+A₂+A₃+A₄ ) (h₁+h₂+h₃+h₄ ) (in cubic units).

Correct answer

240

Step-by-step solution

Volume (V)= 1 3 A₁ h₁ h₁= 3 V A₁ Similarly h ₂= 3 ~V ~A ₂ , ~h ₃= 3 ~V ~A ₃ and h ₄= 3 ~V ~A ₄ S o (A₁+A₂+A₃+A₄ ) (h₁+h₂+h₃+h₄ )= (A₁+A₂+A₃+A₄ ) ( 3 V A₁ + 3 V A₂ + 3 V A₃ + 3 V A₄ )=3 V (A₁+A₂+A₃+A₄ ) ( 1 A₁ + 1 A₂ + 1 A₃ + 1 A₄ ) 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 4 ( 1 ~A ₁ + 1 ~A ₂ + 1 ~A ₃ + 1 ~A ₄ ) ( A ₁+ A ₂+ A ₃+ A ₄ ) ( 1 ~A ₁ + 1 ~A ₂ + 1 ~A ₃ + 1 ~A ₄ ) 16 Hence the minimum value of ( A ₁+ A ₂+ A ₃+ A ₄ ) ( h ₁+ h ₂+ h ₃+ h ₄ )=3 ~V (16)=48 ~V =48 5=240

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