A box with a square base and open top must have a volume of 62,500 cm 3 . Find t

Question

A box with a square base and open top must have a volume of 62,500 cm3. Find the dimensions of the box that minimize the amount of material used.

Sides of Base: __________

Height: __________

NOTE: Keep in mind that the volume of an open box with height h and square base of sidelength a is

V = a2h,

and the area of its surface is

A = a2 + 4ah.

Find a relationship between a and h, using the fact that the volume is a constant. Rewrite the surface area as a function of one variable. Use calculus to find the edge and the height such that the surface area is minimized.

Please show work. Find sides of base & height. THANK YOU!!!

Explanation / Answer

The volume of the required box = 62500 cm^3

Given is square base so the lengeth and breadth of the base are same =a
and let the height = h

So a^2 h = 62500 .... from here we get h = 62500/a^2......................(1)

And the total material going to be used = total surface area of the box.
S = a^2 + 4ah ( it is topless so only the base's area (a^2) will count.)

S = a^2 + 4a ( 62500/a^2) .............( from (1))

S = a^2 + 250000/a

Now we have S as function of one variable and we have to mnimize it.
   dS/da = 2a - 250000/a^2
   For finding its critical points dS/da = 0
   a = 50
   We have to check whether this a actually minimizes the S or not.
By first derivative test (dS/da)50-h <0 and (dS/da)50+h  >0
   Now we can say for sure that the function actually has a relative minima at a =50.

So sides ( length and breadth) = 50 and h = 62500/50^2 = 25 .............................. ( form (1) )

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