Question

A solid rectangular brick is to be made from 1 cubic feet of clay. The length of the brick
must be 3 times as long as its width. The width of brick for which it will have minimum
surface area is a, then a

Answer 1

$b^{3}$Answer:

The volume of rectangular brick will be same as volume of clay.

Let l,b,h be the length, width and height respectively.

Now,

l = 3b

We have,

vol = l x b x h

= 3b x b x h = 1

h = 1/3$b^{2}$

Now,

surface area = 2 l x b + 2 l x h + 2 b x h

                     = 6 $b^{2}$  + 2/b + 2/3b

                     = 6$b^{2}$  + 8/3b

Now, to minimize the surface the area ,

d(SA)/db = 0

=> 12b - 8/3 x 1/$b^{2}$  = 0

=> $b^{3}$ = 8/36 => $b^{2}$ = 2/9

Now the surface area = a = 6$b^{2}$  + 8/3b

a = 18  + 8/3b

= 9/(2/9) ^1/3

Now, $a^{3}$ = 69/2/9 = 32 x 9

$a^{3}$ = 288