Optimization What dimensions should we use for a rectangular

Optimization

What dimensions should we use for a rectangular aquarium of volume $12,000 cm3 if we want to minimize its cost, given that the base costs 5 times more per unit area than the sides. (There is no lid on the aquarium.)

Please help me out with this one. Thanks

Solution

Let the sides of cuboidal Aquarium be a , b and c . Such that a and b are the sides of the base and c is the height

Volume of the Aquarium , V = abc = 12000

Surface Area of the Aquarium , S = ( ab + 2bc + 2ca )

Cost of constructing the Aquarium , C = 5( ab ) + 2bc + 2ca

Making an equation from the concept of Lagrange Multipliers

L = ( abc - 12000 ) + ( 5ab + 2bc + 2ca )

Partial differentiating with respect to a

L/da = bc + ( 5b + 2c ) = 0 => = -bc / (5b + 2c)

Partial differentiating with respect to b

L/db = ac + ( 5a + 2c ) = 0 => = -ac / (5a + 2c)

Partial differentiating with respect to c

L/dc = ab + ( 2b + 2a ) = 0 => = -ab / (2b + 2a)

Therefore ,

Case 1

=> -bc / (5b + 2c) = -ac / (5a + 2c)

=> b( 5a + 2c ) = a( 5b + 2c )

=> 5ab + 2bc = 5ab + 2ac

=> a = b

Case 2

=> -ac / (5a + 2c) = -ab / (2b + 2a)

=> c / (5a + 2c) = b / (2b + 2a)

=> c( 2b + 2a ) = b( 5a + 2c )

=> 2bc + 2ac = 5ab + 2bc

=> 2ac = 5ab

=> 2c = 5b

Combining both

2.5a = 2.5b = c

=> V = abc = a² . 2.5a = 12000

=> a³ = 4800

=> a = (4800)^1/3

=> b = (4800)^1/3

=> c = (2.5)(4800)^1/3