What dimensions minimize the surface area of a box with a fixed volume?

In summary, using the given information and equations, the dimensions that minimize the surface area of a box with a fixed volume are: length = ((9V)/64)^(1/3), width = 8((9V)/64)^(1/3), and height = V/(8(((9V)/64)^(1/3))^2). This solution was found by setting the derivative of the surface area equation equal to zero and solving for x, which gives the critical point for the minimum surface area. The simplification of 64^(1/3) to 4 helps to reduce the answers and make the solution look cleaner.
  • #1
UMich1344
27
0
[SOLVED] Minimizing the Surface Area

Homework Statement



A box has a bottom with one edge 8 times as long as the other. If the box has no top and the volume is fixed at V, what dimensions minimize the surface area?

Homework Equations



V = lwh

SA (with no top) = lw + 2lh + 2wh

The Attempt at a Solution



l = x
w = 8x
h = V/(8x^2)

Finding an equation for the surface area.

SA = lw + 2lh + 2wh
SA = x(8x) + 2x(V/(8x^2)) + 2(8x)(V/(8x^2))
SA = 8x^2 + V/(4x) + 2V/x

Finding the derivative of the equation in order to set it equal to zero to find the critical points, so the minimum can be found.

(d SA)/(d x) = 16x - V/(4x^2) - 2V/(x^2)
(d SA)/(d x) = (64x^3 - 9V) / (4x^2)

(d SA)/(d x) = 0
(64x^3 - 9V) / (4x^2) = 0
(64x^3 - 9V) = 0
64x^3 = 9V
x^3 = (9V)/64
x = ((9V)/64)^(1/3)

Plugging the solution into the equations for the dimensions.

l = x = ((9V)/64)^(1/3)
w = 8x = 8((9V)/64)^(1/3)
h = V/(8(((9V)/64)^(1/3))^2)



I am unsure if I did the right steps in order to find the solution.
Also, I am not very confident in the work I did for each step.
 
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  • #2
It looks good to me, except you can simplify 64^(1/3) into 4 so that you can reduce your answers and it looks cleaner.
 
  • #3
Thanks a lot for looking over it. I just put it into the online grader and it is, in fact, correct.
 
  • #4
I appreciate it.
 

1. What is meant by "minimizing the surface area"?

Minimizing the surface area refers to reducing the amount of exposed surface on an object or material. This can be achieved by decreasing the length, width, or height of the object, or by changing its shape or structure.

2. Why is minimizing surface area important in science?

Minimizing surface area is important in science because it can affect the properties and behavior of materials. For example, a smaller surface area can reduce the rate of heat transfer, which is important in insulation materials. It can also reduce the amount of material needed for a certain function, making processes more efficient and cost-effective.

3. What are some applications of minimizing surface area?

Minimizing surface area has various applications in different fields of science. In chemistry, it is important in controlling the rate of chemical reactions. In biology, it can affect the absorption and transport of nutrients in cells. In engineering, it is crucial in designing structures that are strong and lightweight. It also has applications in nanotechnology, where manipulating surface area can lead to new properties and functions in materials.

4. How is surface area calculated?

The surface area of an object can be calculated by finding the sum of the areas of all its exposed sides. The specific method of calculation will depend on the shape of the object. For example, the surface area of a cube can be found by multiplying the length of one side by itself three times, while the surface area of a cylinder can be found by adding the area of the circular base to the area of the curved side.

5. What are some techniques for minimizing surface area?

There are various techniques that can be used to minimize surface area, including changing the shape of an object to a more compact form, using materials with lower surface tension, and creating structures with micro or nano features. Other techniques include folding, crumpling, and adding coatings or layers to reduce the exposed surface. Computer simulations and mathematical models are also commonly used to optimize surface area in complex systems.

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