Ask Me Help Desk - Lagrange Multipliers (Cal III)

We want to minimize

f(a,b,c)=a+b+c

subject to the constraint:

$a^{2}+b^{2}+c^{2}=10$

Thus, with $g(a,b,c)=a^{2}+b^{2}+c^{2}$ , we have:

${\nabla}f(a,b,c)={\lambda}{\nabla}g(a,b,c)$

$i+j+k={\lambda}(2ai+2bj+2ck)$

Equating like terms leads us to:

$1=2a{\lambda}$

$1=2b{\lambda}$

$1=2c{\lambda}$

So, $\frac{1}{2a}={\lambda}$

$\frac{1}{2b}={\lambda}$

$\frac{1}{2c}={\lambda}$

$\frac{1}{2a}=\frac{1}{2b}=\frac{1}{2c}$

$a=b=c$

sub this into the constraint and we get:

$3a^{2}=10$

$a=b=c=\sqrt{\frac{10}{3}}$

The minimum perimeter is

$3\cdot \sqrt{\frac{10}{3}}=\sqrt{30}\approx 5.48$