the base of an aquarium with given volume V is made of slate and the sides are made of glass. if slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.
The Attempt at a Solution
so this problem fall under finding max, min and saddle points of a function
i think that lagrange multipliers can also be used but id like to first solve it by finding critical points and then the minimum
so first i know that V=xyz and and the total area of the glass will be 2xz+2yz while the area of the base slate will be xy
then the cost function will be C(x,y,z)=the cost of the slate + the cost of the glass
so then C(x,y,z)=5(2xz+2yz)+(2xz+2yz)= 12xz+12yz=z(12x+12y)(?)
im not really sure what to do next, but what i though of doing was to solve for z in V=xyz, then using that z by replacing it for the z in C(x,y,z) then i would only have a function of 2 variables C(x,y) and then i can just go ahead and find partial derivatives and use the second derivative test to find the minimum
is this the correct way to go about finding the dimensions?