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Question

A box with a square base and open top must have a volume of 32000cm3.

How do you find the dimensions of the box that minimize the amount of material used?


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Solution

Minimizing the amount of material used to make an open-top square-base box:

Step-1: Finding the equations of volume and surface area.

Consider the volume of the box of base a×a and height h as, V=a2h.

The amount of surface area has to be minimized to minimize the material used.

The surface area is given by, A=a2+4ah.

Substituting for h from the first equation; we get, A=a2+4aVa2

Volume is given as 32000cm3; substituting it, we get, A=a2+4×32000a:

A=a2+128000a

Step-2: Finding the critical point by finding the first derivative and equating it to zero.

Differentiating A with respect to a; we get,

dAda=2a-128000a2

Equating the first derivative to zero; get the critical point as,

2a-128000a2=02a=128000a2a3=64000a=640003a=40

Step-3: Checking that the critical point gives the second derivative at that point as positive.

The second derivative of A with respect to a, we get,

d2Ada2=2-2×128000a3d2Ada2=2+256000a3

Now,

A''40=2+256000403=2+25600064000=2+4=6

At a=40, we get the second derivative as positive. Hence, the area is minimum.

Step-4: Finding the height of the box.

We get the height from the first equation as,

h=32000a2=32000402=320001600=20

So, the dimensions of the box are: 40cm×40cm×20cm

Therefore, the required dimensions of the base of the box is 40cm×40cm and height, 20cm.


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