1. ## Lagrange Multipliers

The base of an aquarium in the shape of rectangular box with a given volume V is made of slate and the four sides are made of glass. If slate costs five times as much as glass per unit area, find the dimension of the aquarium that minimize the cost of the materials.

2. Originally Posted by CandyKanro
The base of an aquarium in the shape of rectangular box with a given volume V is made of slate and the four sides are made of glass. If slate costs five times as much as glass per unit area, find the dimension of the aquarium that minimize the cost of the materials.

Let the dimensions of the box be $\displaystyle x,\ y,\ z$ , then the cost is proportionsl to:

$\displaystyle f(x,y,z)=5 xy + 2xz + 2yz$

and we have the constraint:

$\displaystyle g(x,y,z)=xyz$

with $\displaystyle g(x,y,z)=V$.

Then our Lagrangian is:

$\displaystyle \Lambda(x,y,z,\lambda)=f(x,y,z)-\lambda(g(x,y,z)-V)$

Now you need to find the stationary points of $\displaystyle \Lambda$ to find the candidates for the maximising solution.

.