A box with a square base and no top must have a volume of 10 000 cm^3. If the smallest dimension is 5 cm, determine the dimensions of the box...

Let the sides of the base be x and the height be
h.

Then the volume is given
by

v= x^2 * h .

==>
10,000 = x^2 h.............(1)

We need to find the minimum
surface area of the box.

We know that SA = area of the base
+ area of the sides.

==> SA = x^2 +
4(xh)

==> SA = x^2 +
4xh

But we know that h=
10000/x^2

==> SA = x^2 +
4x(10000/x^2)

==> SA = x^2 +
40,000/x

==> SA = (x^3 +
40,000)/x

Now we need to find the first derivative in order
to find the minimum value of SA.

==> SA' = (3x^2)(x)
- (x^3+4,000) / x^2

==> SA' = ( 3x^3 - x^3 -
40,000)/x^2 = 0

==> 2x^3 =
40,000

==> x^3 =
20,000

==> x = 27.14 (
nearly)

==> h= 10,000/x^2 = 10,000/736.081 =
13.57

Then the dimensions for the base is x=
27.14 and the height is h= 13.57 to have the maximum surface
area.