Question

An aquarium is to be built to hold 60 m3of volume. The base is to be made of slate and the sides aremade of glass, and it has no top. If stone costs $120/m2and glass costs $30/m2, find the dimensions which willminimize the cost of building the aquarium, and find the minimum cost.

Answer 1

Aquarium dimensions:

x = 3,106 m

h = 6,22 m

C(min) = 1277,62 $

Step-by-step explanation: (INCOMPLETE QUESTION)

We have to assume:

The shape of the aquarium (square base)

Let´s call "x" the side of the base, then h ( the heigh)

V(a) = x²*h h = V(a)/x²

Cost of Aquarium C(a) = cost of the base (in stones) + 4* cost of one side (in glass)

C(a) = Area of the base *120 + 4*Area of one side*30

Area of the base is x²

Area of one side is x*h or x*V(a)/x²

Area of one side is V(a)/x

C(x) = 120*x² + 4*30*60/x

C(x) = 120*x² + 7200/x

Taking derivatives on both sides of the equation we get

C´(x) = 2*120*x - 7200/x²

C´(x) = 0 means 240 *x - 7200/x² = 0

240*x³ - 7200 = 0

x³ = 7200/240

x = 3,106 m and h = 60 /x² h = 6,22 m

and C (min) = 120*(3,106)³ - 7200 / 3,106

C(min) = 3595,72 - 2318,1

C(min) = 1277,62