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 (Aquarium has rectangular base).

x = ∛ 2*V/5

y = ∛ 2*V/5

h = V / ∛ 4*V²/25

Step-by-step explanation:

Dimensions of the aquarium base is x*y

We call c₁ cost per unit area of the sides, then cost per unit area of slate is equal 5c₁.

let call h the height of the aquarium then volume of the aquarium is:

V = x*y*h where h = V / x*y

As the base is a rectangular one there are 2 sides x*h. and 2 sides y*h

According to this:

Ct (cost of aquarium) = cost of the base + cost of the sides

cₐ (cost of the base) = 5*c₁*x*y

c₆ (cost of the sides) = c₁*2*x*h + c₁*2*y*h

C (t) = 5*c₁*x*y + 2 * c₁*x * V/x*y + 2 * c₁*y * V/x*y or

C (t) = 5*c₁*x*y + 2*c₁*V/y * 2*c₁ * V/x

Taking partial derivatives en x and y we have:

C' (x) = 5*c₁*y - 2*c₁*V/x²

C' (y) = 5*c₁*x - 2*c₁*V/y²

C' (x) = C' (y) ⇒ 5*c₁*y - 2*c₁*V/x² = 5*c₁*x - 2*c₁*V/y²

or 5*y - 2*V/x² = 5*x - 2*V/y²

(5*y*x² - 2*V) / x² = (5*y²x - 2*V) / y²

(5*y*x² - 2*V) * y² = (5*y²x - 2*V) * x²

5*y³*x² - 2*V*y² = 5*y²x³ - 2*V*x²

5*y³*x² - 5*y²x³ = 2*V * (y² - x²)

by symmetry x = y

Then using x = y and plugging that value on the derivatives

C' (x) = 5*c₁*y - 2*c₁*V/x²

C' (x) = 5*c₁*x - 2*c₁*V/x²

C' (x) = 0 ⇒ 5*c₁*x - 2*c₁*V/x² = 0

5*x - 2*V/x² = 0 ⇒ 5*x³ - 2*V = 0 ⇒ 5*x³ = 2*V ⇒ x³ = 2*V/5

x = ∛ 2*V/5 and y = ∛ 2*V/5 and h = V / x*y h = V / ∛ 4*V²/25