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I want to minimize the surface area of a rectangular prism, with a constant volume. The dimensions of the prism is $25 \text{ cm} \times 8 \text{ cm} \times 8 \text{ cm}$. If it is "flattened out" [with the top face not included], then the dimensions of the "flattened out" prism is $41\text{ cm} \times 24 \text{ cm}$.
The volume and surface area will be functions of three variables, length, width and height, one of which can be expressed in terms of the other two. So one must optimize a function of two variables subject to a constraint, suggesting Lagrange multipliers. $\endgroup$
$\begingroup$ A cube has the smallest surface:volume ratio, so the expected answer is $\sqrt[3]{50}m$. Of course, to prove this, you can write the dimensions of the rectangular prism in terms of one variable and the volume, then use your typical optimization techniques to proceed. $\endgroup$ –
For example, a surface area of 18 can have volume 4 (4x1x1), or volume 8 (2x2x2), or any other of infinitely many possibilities. You don't have enough information to solve the problem. However, if you have additional information, like integer dimensions, then it is possible (although there may be multiple solutions).
So, for each one you'd calculate the area of the bottom, then the area of the outer sides. For the circular one, imagine slicing the tube down the side and unrolling it. It would have an area of the height times the circumference of the bottom circle. Likewise the area of the other is the area of the square bottom, plus the area of the four walls.
Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube.
Here is the question: A rectangular prism has a volume of $720$ cm $^3$ and a surface area of $666$ cm $^2$.If the lengths of all its edges are integers, what is the length of the longest edge?
1) The area of a unit square is 1; 2) Congruent polygons have equal areas; 3) If a simple polygon P is decomposed into a finite number of parts, then the area of P is the sum of the areas of the parts. Theorem: The area of a rectangle is the product of the lengths of its sides. We will divide it into 3 cases: 1) Both lengths are whole numbers;
$\begingroup$ I mean that the numerical value of the volume and surface area is the same. I know that the edges have lengths 3cm, 7cm and 42cm which gives a volume of 882cm cubed and a surface area of 882cm squared, but I need to find an algebraic way to work out the unknown lengths. $\endgroup$ –
The surface area of B, the rectangular prism's width side, will also be simple as it is $23 * 6.33...$ which equals $145.66...$. Again multiply by two to account for the other side resulting in: $$291.3333...$$. To find the surface of C, the triangular prism's side, we will need to multiply the length, 96, by the height of the side.