0
Given the perimeter of the box and the total surface area what is the maximum volume of the box?
Wiki User
∙ 12y ago
With a fixed surface area you can actually maximize the volume of the box without using multi-variable calculus. Here's how:
Suppose the dimensions are x, y, z with 2(xy + yz + zx) = C for some constant C. This implies xy + yz + zx = C1 for a different constant C1. We will use what mathematicians call the AM-GM inequality (Arithmetic mean - geometric mean inequality) which says that, for some positive numbers, the arithmetic mean is always greater than or equal to the geometric mean, with equality occurring iff all the numbers are equal.
The AM-GM inequality says that
(xy + yz + zx)/3 >= (xy*yz*zx)1/3 (couldn't do cube root)
((xy + yz + zx)/3)3/2 >= xyz
The left side is equal to a constant (since it has the expression for surface area), so the maximal value of xyz (the volume) is equal to that constant. This happens when x = y = z, i.e. the box is cube-shaped.
Wiki User
∙ 12y ago
Add your answer:
Earn +20 pts
How are area and perimeter and volume related?
They are characteristics of geometric shapes. However, there is no simple relationship. A rectangle with a given perimeter can have a whole range of areas.
How to Maximize volume given surface area?
make it spherical
How to find the surface area and volume of a sphere with a given radius.?
Given a sphere of radius r, Surface area = 4{pi}r2 Volume = (4/3){pi}r3
What is the maximum area of a rectangle that can be made with a perimeter of 68 inches?
The maximum area with straight sides and a given perimeter is a square.The sides of the square are (68/4) = 17 inches.The area is (17 x 17) = 289 square inches
How do you find height in a cylinder given surface and volume?
By dividing its cross-section area into its volume
What is the maximum area for a rectangle whose perimeter is 136?
The maximum area for a rectangle of fixed perimeter is that of the square that can be formed with the given perimeter. 136/4 = 34, so that the side of such a square will be 34 and its area 342 = 1156.
How are area and perimeter and volume related?
They are characteristics of geometric shapes. However, there is no simple relationship. A rectangle with a given perimeter can have a whole range of areas.
Why do alveoli have folds in them?
increase surface area for a given volume
How to Maximize volume given surface area?
make it spherical
How to find the surface area and volume of a sphere with a given radius.?
Given a sphere of radius r, Surface area = 4{pi}r2 Volume = (4/3){pi}r3
How do you find height in a cylinder given surface and volume?
By dividing its cross-section area into its volume
What is the maximum area of a rectangle that can be made with a perimeter of 68 inches?
The maximum area with straight sides and a given perimeter is a square.The sides of the square are (68/4) = 17 inches.The area is (17 x 17) = 289 square inches
What is the formula for the volume of a sphere in terms of surface area S?
Given the surface area, where S=surface area, the formula for finding the volume isV = √(S / 4pi)
Why is a bubble spherical?
The spherical shape is the smallest surface area for a given volume. This comes about naturally when a surface under pure surface tension contains a fluid volume.
Area is the amont of space in a given box?
That's volume. Area is the measurement of a given surface.
How is area perimeter and volume related?
In general the larger the perimeter (of a flat shape) the greater the area. Given two congruent shapes the one with the larger perimeter has a greater area.But two shapes that are not congruent (or almost so) do not follow this rule: for example a rectangle fifteen units long and one unit wide has an area of 15 square units and a perimeter of 32 units. While a square with edges four units has an area of sixteen square units (one more than the other rectangle) but a perimeter of only sixteen units (half that of the long thin rectangle).So too with surface area and volume. Of two congruent 3 dimensional shapes, the one with the larger volume will also have a larger surface area.
What geometric shape has the smallest surface area for a given volume?
A sphere
