SOLUTION: A box with a square base and open Top is to have a volume of 62.5 cm³. Neglect the thickness of material used. Find the dimension that will minimise the amount of material used an

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Question 1149783: A box with a square base and open Top is to have a volume of 62.5 cm³. Neglect the thickness of material used. Find the dimension that will minimise the amount of material used and hence find the surface area of the material used to make the Box.
Answer by ikleyn(52803) About Me  (Show Source):
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If x is the side length of the base square and h is the height of the box, then the area of the open box is

    A(x,h) = x%5E2 + 4xh.    (1)


The volume of the box is

    V(x,h) = x%5E2%2Ah.        (2)


We need to minimize the surface area (1) under the restriction

    x%5E2%2Ah = 62.5  cm^3.     (3)


From (3), express  h = 62.5%2Fx%5E2 and substitute it into (1). You will get then

    A(x) = x%5E2 + %284%2A62.5%2Ax%29%2Fx%5E2 = x%5E2 + 250%2Fx.    (4).


Thus we need to minimize function A(x) of one variable "x" expressed by (4).


For it, take the derivative


    A'(x) = 2x - 250%2Fx%5E2


and equate it to 0 (zero).  You will get


    2x - 250%2Fx%5E2 = 0,

    2x^3 - 250 = 0,

    x^3 = 250/2 = 125,

    x = root%283%2C125%29 = 5  cm.


Then  h = 62.5%2Fx%5E2 = 62.5%2F5%5E2 = 2.5 cm.


Thus the optimal dimensions are  x = 5 cm;  h = 2.5 cm.                                     ANSWER

The surface area of the open box is  A = x%5E2 + 4*x*h = 5%5E2 + 4*5*2.5 = 25 + 50 = 75 cm^2.    ANSWER


Partial CHECK.  The volume = x%5E2%2Ah = 5%5E2%2A2.5 = 62.5 cm^3.    ! Correct !

Solved.