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What I am asking is that you say that a rectangular prism with a known sum of sides has the least surface area when it is a cube. But, a cube with 6 has the maximum possible surface area. Why is it that a rectangular prism with known sum of sides has largest possible surface area when it is cube, but a square based pyramid with known volume has ...
Sorted by: 2. If the question is to find dimensions of a rectangular box, with constant volume of 1600, that minimizes the surface area of the box. The box is closed on top. Surface area S = 2lw + 2lh + 2wh. V = 1600 = lwh h = 1600 wl. So, S = 2lw + 3200 w + 3200 l. Now take partial derivative wrt w and l and equate to zero to find w, l, h that ...
I confirmed that to obtain the max volume, the prism would be a cube. So in this case, we would try to find the volume assuming that the prism is a cube. 1 Face of the cube would be 300/6=50. 1 Side of the cube would be √50≈7.07. The volume would be 7.07 * 50 = 353.5.
$\begingroup$ 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. $\endgroup$ –
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. $\endgroup$
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).
2. we will keep the volume at 1. let the base have length x and width y. then the volume constraint makes the height of the box 1 xy. you need to minimize the surface area A = 2(xy + 1 x + 1 y), x> 0, y> 0 now you can use the am-gm inequality a + b + c 3 ≥ (abc)1 / 3 to show that A ≥ 6(xy1 x1 y)1 / 3 = 6. therefore the minimum surface area ...
1 Answer. Sorted by: 1. If we cut equal-sized square notches of side length x x from the corners of the sheet, then the volume of a rectangular prism from the sheet would be. V(x) = x(a − 2x)(b − 2x) = 4x3 − 2(a + b)x2 + abx V (x) = x (a − 2 x) (b − 2 x) = 4 x 3 − 2 (a + b) x 2 + a b x. where a a and b b are the lengths of the sides ...
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.
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?