Find the minimum value of V among all planes passing through the point P = (5, 5, 5) .

Question

A plane with the equation x/a + y/b + z/c = 1(a, b, c > 0) together with the positive coordinate planes forms a tetrahedron of volume V = 1/6abc. Find the minimum value of V among all planes passing through the point P = (5, 5, 5) .

Answer 1

We have V = 1/6abc under constrains 5/a + 5/b + 5/c = 1.We applying Lagrange Multipliers method

F(a,b,c) = 1/6abc - λ(5/a + 5/b + 5/c - 1); F_a = 1/6bc + 5λ/a² = 0; F_b = 1/6ac + 5λ/b² = 0; F_c = 1/6ab + 5λ/c² = 0 and 5/a + 5/b + 5/c = 1. We have system of 4 equations with 4 variables a, b, c and λ

We have a²bc = b²ac = c²ab = - 30λ, from here a = b = c.

5/a + 5/a + 5/a = 1; a = 15, b = 15 and c = 15

V_min = 1/6·15³ = 562.5

To show that this is minimum, let take any different plane passing this point (5, 5, 5).For example:

x/20 + x/20 + x/10 = 1. so v = 1/6·20·20·10 = 6662/3 > 562.5