two similar solids have volumes of 27cm3 and 8 cm3. find the ratio fo their surface areas... answer in steps?
475972
The volume of a cylinder is given by\n\nV = pi*r^2*h\n\nAnd the surface area of a cylinder is given by\n\nSA = 2*pi*r^2 + 2*pi*r*h\n= 2*pi*r(r + h)\n\nNow, we don't have enough information to specifically find either the radii or heights - we only have a volume.  But we can still work with that based on the fact that the cylinders are similar.\n\nIf the cylinders are similar, their measurements are proportional.  In other words, the radii are proportional and the heights are proportional (r1/r2 = h1/h2 and r1/h1 = r2/h2).  So let's just assume that the larger cylinder has a radius of 3 and a height of 3 (plug in to the volume formula and it works out to 27*pi cm^3, but for the sake of this argument, we'll ignore the pi for now).  For the smaller cylinder to be similar, it would have to have a radius of 2 and a height of 2 (again ignoring pi).\n\nThe surface area of the larger cylinder is then\n\nSA = 2*pi*r(r + h)\n= 2*pi*3(3 + 3)\n= 36*pi\n\nAnd the smaller one would have a surface area of\nSA = 2*pi*r(r + h)\n= 2*pi*2(2 + 2)\n= 16*pi\n\nSo your ratio would then be 36*pi / 16*pi = 9/4.  (If we went back and corrected out radii and heights for pi, they would have all cancelled out by this step.)\n\nNow, what if my initial assumption was wrong and the radii and heights weren't 2 and 3?  Well, we can still work with that, but we'd end up with the same ratio of 9/4.  If the smaller cylinder had a height of 4, then the radius would have to be sqrt(2) for the volume to work out to be 8.  So the ratio of radius to height would be sqrt(2)/4.  Because the two cylinders are similar, this ratio must hold for the larger one.\n\n(Crazy math warning!)\nSo r/h = sqrt(2)/4, so r = h*sqrt(2)/4.  We can find the radius and height of the larger cylinder now:\n\nV = pi*r^2*h\n27 = pi*(h*sqrt(2)/4)^2*h\n27 = pi*(h^2/8)*h\n27 = (pi*h^3)/8\n216 = pi*h^3   <-  we'll drop the pi at this point - same reason as before\n216 = h^3\nh = 6\n\nr = h*sqrt(2)/4\n= 6*sqrt(2)/4\n= 3*sqrt(2)/2\n\nNow let's compare surface areas of the two:\n\nSA(big) = 2*pi*r(r + h)\n= 2*pi*(3*sqrt(2)/2)(3*sqrt(2)/2 + 6)\n= (3*pi*sqrt(2))(3*sqrt(2)/2 + 6)\n= 9*pi + 18*pi*sqrt(2)\n= 9*pi(1 + 2sqrt(2))\n\nSA(small) = 2*pi*r(r + h)\n= 2*pi*(sqrt(2))(sqrt(2) + 4)\n= 4*pi + 8*pi*sqrt(2)\n= 4*pi(1 + 2sqrt(2))\n\nAnd if you took this ratio of big to small, you have 9*pi(1+2sqrt(2)) / 4*pi(1+2sqrt(2)) = 9/4!\n\nSo my original assumption is right, and the ratio turns out to be 9/4 for these two cylinders.\n\nNow I used cylinders for this (I have no idea why), but it turns out that it works for any solid.  There's a relationship among these measurements - if we compare cubes:\n\nRatio of a side = s1/s2\nRatio of area = (s1)^2/(s2)^2\nRatio of volume = (s1)^3/(s2)^3\n\nSo for our case, the ratio of the area can be found by taking the cube root of the ratio of volumes, and then squaring it - 9/4.
895037
surface area of a cube is 6a^2\nvolume=a^3\ntheir relation: V/SA = a/6\n\nif its cut into half, volume will be:\n(a/2) x a^2 = (a^3) / 2\nthe surface area will be 2a^2 + 4a(a/2) = 4a^2\nnow their relation: V/SA = a/8\n\nnote: crazy horse's answer went wrong when he assumed a halved cube has only two of its sides halved. in fact, it has its 4 sides halved.
208438
There is a delightful article by the late Dr. John Kenny on that, back a few years ago, looking at the formulae for surface area and volumes of spheres; I'll see if I can find it for you--it's in my files SOMEWHERE--and give you the reference, and the formulae he generated. It turns out that they're not precisely the same as what you've got--but not terribly far from it.\n\nAs I said, I'll see if I can find it, and I'll get you the reference and the formulae.\n\nAbout 9pm CST...\n\nOK, here is the formulae, done the best I can on this thing...\n\n\nFor a sphere of dimension 2*k, radius R\n\nVolume: (pi^k)*(R^(2*k))/k!\n\nFor a sphere of dimension 2*k + 1 and radius R\n\nVolume:\n(2^(2*k+1))*k!*(pi^k)* (R^(2*k+1))/(2*k+1)!\n\n\nTo get the surface areas for each of these, all you do is differentiate with respect to R...\n\nAs I said, it was a little more complex than you thought--the surface areas come out a little bit differently for odd and even numbered dimensions.  I do not recall why, frankly.
728936
1. obtain formulas for surface area and volume of cylinder\n2. let volume equal 1000 and solve this volume equation for h\n3. substitute the expression from previous step into surface area    equation to get equation with only r's\n4. differentiate surface area formula with respect to r set this equal to 0 and solve for r i got   r=10/3rd root(pi)\n5. substitute value for r into volume equation solve for h  i got h=10/(pi)^(2/3)\n\nincidentally, i used 2nd derivative test to conclude r is minimum\n\n\n\ni'll get back to u with more
801553
lol.\nfind formula for area of spiral.  subtract smaller from bigger.  sry no more help.
335828
The radius is half the diameter.\nThe diameter is twice the radius.\nd = 2r\n8 = 2r\nr = 8/2\nr = 4 inches
84844
Height = H = 10"\nRadius = r = 4"\n\nVOLUME (not area, volume is 3-dimensional) = (π/3)(r^2*H) = (160/3)*π\n\nSURFACE AREA = (πr)(sqrt(r^2+H^2)) = 4(sqrt(116))*pi
407730
Let's look at the general case. It turns out that for even dimensional spheres, the ratio of the 'surface area' to the 'volume' of the next lower dimensional sphere is irrational in even dimensions and rational in odd dimensions. Your cases are just special cases of this general result.
937032
I'm assuming:\nThe box has a square top and bottom.\nThe volume v is fixed.\nr is the length of a side of the top.\nh is the height of the box.\n\nv = r^2*h\nh = v/r^2\n\nThe surface area is a top and bottom both r x r and\nthe sides which are r x h.\n\nArea = 2*r^2 + 4*r*h = 2^r^2 + 4*r(v/r^2) = 2r^2 +4v/r\n\ndA/dr = 4r -4v/r^2\n\nSet to 0 and solve\n0 = 4r -4v/r^2\n\n0 = 4r^3 -4v\n0 = r^3 - v\nv = r^3, or  r = v^(1/3)\nThat means that the container is a cube.
103215
With this given data, it is not possible\n\nalthough the area is totalling to be 70 for both cases (7*10 and two pieces of 8*8 + 1*6) but with one division only, this is not possible\n\nPLEASE DO LET ME KNOW IF THERE IS SOME SOLUTION (i found it very interesting)
486266
the formula for volume of a cylinder is V = h * pi * r^2.\nAlternatively, the surface area is A = h * pi * r * 2.\n\nSo, if you have either the surface area or the volume, you can calculate for the height.\n\nH = Volume / pi * r^2\nH = Area / pi * r * 2
1087751
The amount of air is the difference in volumes.\nA = V(r+1) - V(r) = (4/3)pi (r+1)^3 - (4/3)pi r^3\nA = (4/3)pi((r+1)^3 - r^3)
706195
the sfce area=7 times the volume\nvolume=1/7 th the S.A.\n=1/7*42\n=6cc
329292
V=(4/3)*pi*r^3\n\nwhere r is the radius of the sphere and V is volume.
863802
area=(1/2)d1*d2\n=(1/2)*28*13\n=14*13\n=182 sft
218464
If by area you mean "Surface area" then\nSA(sphere) = 4(pi)(r)^2\nand\nV(sphere) = 4(pi)(r)^3/3\n\nSince SA = 196 cm^2, then\n\n196cm^n = 4(pi)r^2\nr^2 = 49/pi\nr = 7/sqrt(pi) or approx. 3.949 cm\n\nfor the volume,\nV = 4(pi)(r)^3/3\nV = 4(pi)(7/pi^0.5)^3/3\nV = (4pi/3)(343/pi^1.5)\nV = (1372/(3pi^.5)) or approx. 258.02 cm^3\n\n^_^ Hope this helps ^_^
679609
Surface Area=sqrt{3}a^2\n    Volume =(1/12)sqrt(2)a^3\nhttp://en.wikipedia.org/wiki/Tetrahedron