Question

a cuboidal shaped Godown with square base is to be connected three times as much cost per square metre is incurred for constructing the roof as compared to the walls. find the dimensions of the Godown if it is to enclose a given volume and minimise the cost of constructing the roof and the walls.​

Answer 1

Answer:

Step-by-step explanation:

Length of the godown = breadth of the godown =  l = ∛(2v/3)

Height of the godown = h = ∛(9v/4)

Where v is the volume of the godown.

Step-by-step explanation:

Since it has a square base,

Let the length = breadth = l

Let the height = h

Let the constant volume be = v

Hence,

v= l x l x h = l²h

=> h = v/l²   .............eq 1

Now it is given that the cost of constructing the roof is 3 times cost of constructing the walls,

let the cost of construction of the wall = Rs r

So cost of construction of roof = 3r

Area of 4 walls = 4lh = 4 x l x v/l² = 4v/l

=> cost of construction of 4 walls = 4vr/l

Area of roof = l x l = l²

=> cost of construction of roof = 3rl²

Hence total cost of construction of godown = C =   3rl² +  4vr/l

We know that for a minima to occur its differential should be 0.

Hence differentiating the above cost w. r. t l (as l is the variable we get),

dC/dl = 6rl - 4vr/ l² = 0

=> 6rl = 4vr/l²

=> l³ = 2v/3

=> l = ∛(2v/3)

puttingthe value of l in eq 1 we get,

h = v/l²

= v/∛(2v/3)²

h = ∛(9v/4)