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.
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.