Yefim S. answered • 10/26/20

Math Tutor with Experience

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/a^{2}= 0; F_{b}= 1/6ac + 5λ/b^{2}= 0; F_{c}= 1/6ab + 5λ/c^{2}= 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 a^{2}bc = b^{2}ac = c^{2}ab = - 30λ, from here a = b = c.

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

V_{min}= 1/6·15^{3}= 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