# 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 that...minimize the amount of material used. ...

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 that...

minimize the amount of material used.

Answer in textbook is 27.14 cm by 27.14 cm for the base and height 13.57 cm.

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### 1 Answer

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.**