The volume of a square-based rectangular cardboard box is to be 1000 cm3. Find the dimensions so that the quan

Question

The volume of a square-based rectangular cardboard box is to be 1000 cm3. Find the dimensions so that the quantity of material used to manufacture all 6 faces is a minumum. Assume that there will be no waste material. The machinery avaiable cannot fabricate material smaller in lenght than 2 cm.

The answer is 10 cm x 10 cm x 10 cm

Please show your work. Thank you.

Answer 1

a "square-based rectangular box" has length the same as width (to keep the base square). This gives you these equations, with w (width and length) and h (height) of the box:

surface area (square top/bottom, plus four sides):
a = 2w^2 + 4wh
volume:
v = w^2h

You're told that the volume is 1000cc, so:

v = w^2*h
1000 = w^2*h
h = 1000/w^2

Substitute into the area equation:

a = 2w^2 + 4wh
a = 2w^2 + 4w(1000/w^2)
a = 2w^2 + 4000/w

Now to find the optimum, take the derivative and set it equal to zero:

da/dw = 0
d(2w^2 + 4000/w)/dw = 0
4w - 4000/w^2 = 0
4w = 4000/w^2
w^3 = 1000
w = 10cm

Now that you know w, you can solve for h:

1000 = w^2 * h
1000 = 10^2 * h
h = 10cm

Thus the optimum dimensions for 1000cc volume, and minimum surface area is 10x10x10.

Note: This problem is more interesting if the box has no top, or if the sides cost a different amount than the top/bottom and we are minimizing construction cost, etc.

Answer 2

Since the base is a square, we know that two of the dimensions (say length and width) are equal.
Call each x cm.
Let the height = h cm

Then the area of the base is x^2.

We know that the volume is 1000 cm^3, so

1000 = hx^2 Solving this for h, we get
h = 1000/x^2

The surface area (S) is 2x^2 + 4hx. This is what we want to minimize.

By substituting 1000/x^2 in place of h, we get

S = 2x^2 + 4(1000/x^2)x
S = 2x^2 + 4000/x

This is what you need to minimize, so take the derivative and solve for when S' = 0

S' = 4x - 4000/x^2
0 = 4x - 4000/x^2
4000/x^2 = 4x
1000 = x^3
10 = x

Then, since x = 10, solve for h, and you will see that x also is 10 cm, so you end up with the 10 cm x 10 cm x 10 cm cube that you said was the answer.

Hope this helps!

Answer 3

This Site Might Help You.

RE:
The volume of a square-based rectangular cardboard box is to be 1000 cm3. Find the dimensions so that the quan
The volume of a square-based rectangular cardboard box is to be 1000 cm3. Find the dimensions so that the quantity of material used to manufacture all 6 faces is a minumum. Assume that there will be no waste material. The machinery avaiable cannot fabricate material smaller in lenght than 2...

Answer 4

Shape of the rectangular Card board is a CUBOID with a square base.

Volume = Area of the Cross-section * Height cm ^ 3

= Area of the square * height of the Prism = 1000 cubic cm.

=> a ^2 * h = 1000

a can be a multiple of 2 cm & h can be a milyiple of 2.

So The ONLY dimensions possible is 10 * 10 * 10 c.m

In this case the Cardboard bix will be a CUBE & not square based Rectabgular one.

So, the problem is misleading.

If the Box is a CUBE, Everything becomes easy.

V = a ^3 = 1000 cubic cm

a = Cube root of 1000

= Cube root of 10 * 10 * 10.

= 10 c.m.

So the dimensions of the CUBICAL Box = 10 cm * 10 c.m. * 10 c.m.

Answer 5

The solid with the smallest surface area to volume ratio is the sphere. That's why soap bubbles are spherical. Similarly, the rectangular prism that has the smallest surface area to volume is a cube. You need a cube with edge of 10 cm. Each side will be 100 cm2 and the total of all six sides would be 600 cm2.

Answer 6

Volume Of A Rectangle Box