Can anyone solve this maximum volume math problem?
912780
Let x be the length of the corners.  Then the base will be (30 - 2x) by (30 - 2x) and the height will be x.\n\nYou want that to be a maximum, so create a formula for the volume, take the first derivative, set it to zero and you'll have the maximum.\n\nf(x) = x(30 - 2x)(30 - 2x)\nf(x) = x(900 -120x + 4x²)\nf(x) = 4x^3 - 120x² + 900x\n\nNow take the derivative\nf'(x)= 12x² - 240x + 900\n\nNow factor it:\nf'(x) = 12(x² - 20x + 75)\nf'(x) = 12(x - 5)(x - 15)\n\nAnd set it to zero:\n12(x - 5)(x -15) = 0\n\nSo x must be 5 or 15, but at x = 15, you have a minimum volume (0).  At x = 5 you have a maximum volume of 5 in x 20 in x 20 in = 2000 cu. in.
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)
760290
Popcorn ball. . . . \n\nGoogle search:  volume sphere (should have used "volume sphere," but it turned out not to matter)\n\nhttp://www.aaamath.com/geo79-volume-sphere.html\n\n4/3 (pi) r^3\n\n4/3 times pi times (6 to the 3rd power) \n\nequals 4/3 times pi times 216. . . and work from there.\n\nGoogle search:  volume cylinder\n\nhttp://www.321know.com/exp79_x5.htm\n\n(area of the base) * height =(pi r squared times height)\n\nPi times (16 squared) times 275. . . work from there.
231738
to complete the problem:\n7.5o + 3.5b = 4375  Eq.#1\no + b = 650  Eq.#2\n\nSolve Eq.#2 for o:\no = 650 - b  Eq.#3\n\nSubstitute Eq.#3 into Eq.#1:\n7.5(650-b) + 3.5b = 4375\n4875 - 7.5b + 3.5b = 4375\n\nSolve for 'b':\n4875 - 4375 = 7.5b - 3.5b\n500 = 4b\nb = 125\n\nSubstitute b = 125 into Eq. #2:\no + 125 = 650\no = 600-125\no = 525\n\nTherefore, there were 125 balcony seats sold, and 525 orchestra seats sold.
1068729
i think n=6..\nyes this can be done..if you know the formula..\nof volume and surface arae of a hypersphere and also the form of gamma fucntion..\nhope you can do it yourself..i saw these two links and based on the formula given for volume and surface ares and the gamma fucntion expression its evident that n=6.\nHope you can figure it out yourself.\nvolume= pi^3/6 (radius =1 here)
920579
use synthetic  divsion method \ni think this link would help\nhttp://www.purplemath.com/modules/synthdiv.htm
108007
I think it goes like this:\n20% = .2, 25% = .25\n\nlet x = final volume\n.25(12) = .2(x)\n1.25(12) = x\n15 = x\nif x = final volume, volume added = x - 12\nor 15\n15 - 12 = 3\nadd 3 L
974937
please complete the equation
942547
Can you please check you haven't made a mistake somewhere? I have just calculated (with 99% certainty) that there are no solutions to those equations with those restrictions.\n\nEdit - OK, I assumed you made exactly one mistake. By removing each possible word and trying to solve it, I discovered that removing 'SIGH' was the only option that gave an answer.\nt 1\nh 8\na 3\nn 7\nk 6\ns 5\ng 9\ni 4\nv 2\nwhere SIGH = 26, not 25. Then THANKSGIVING = 65.\n\nThat was actually a cooler puzzle that the original, assuming you made one mistake :)
741778
This is not tricky at all. The volume is 1-1/2 cylinders of "height" 50,000 meters and radius 4 meters.\n\nThe formula for the volume of a cylinder is\n\nV = pi * r^2 * h
754188
you divide the mass by the volume\n1.2.14 g/cm^3\n2..82 g/cm^3\ngood luck!
918429
use the formulas your given\nto find the volume of the cylinder\nthen pretend the semi hemi on  the top is a sphere find the volume and multiply by 1/2\n\nadd two totals to find volume
590580
Volume = length x width x height = 3*3*3 = 27 cubic inches
598778
Here's the correct solution.\n\nA cylindrical package with height(length) h and radius r will provide the maximum volume under the restriction that the sum of the girth and length equals a given constant C.\n(In your case C = 108")\n\nGeneral case, for arbitrary C\nFor a cylinder, the girth = 2pi*r, so our restriction is:\n\nC = 2pi*r + h\n\nThe volume V = pi*r^2*h\n\nFirst solve for h in terms of C:\n\nh = C - 2pi*r  Substitute this into volume formula:\n\nGives  V = pi*(r^2)*(C-2pi*r)\n\nV = pi*C*(r^2) - 2(pi^2)r^3\n\ndetermine dV/dr = 2*pi*r*C - 6*(pi^2)r^2\n\nSet dV/dr = 0, implies 6*(pi^2)r^2 = 2*pi*r*C\n\nSo 2pi*r = 2C/3, and therefore h = C/3, since 2pi*r + h= C\n\nSo if C = 108" then for maximum volume h= 108/3 =36\nand 2pi*r = 2C/3 so r = C/3pi and r = 108/3pi = 11.46 \n\nMax Volume = pix(11.46^2)x36 = 14,853 cubic in.\n\nWe can test that this is a maximum volume by testing the neighborhood. for example 3C/10 < C/3 < 3C/8\n\nlet h=3C/8 = 40.5 then r = 67.5/2pi= 10.74\n\nand Volume = pi*(10.74^2)*40.5 = 14,684 cubic in. This is less than the Max volume of 14,853 \n\nIf h= 3x108/10 then Volume = pi(12.03^2)*32.4 = 14,730\n\nThis is again less than Max volume, and difference increases the more h varies from C/3 = 108/3 = 36
907697
You need to find an equation first. try drawing the box out it may help. Draw a rectangle label one side 16 inches the other 30 then draw squares at each corner.\n\nLet x be the length of one of the square's sides\n\nVolume = length * width * height\n\n= (30-2x) * (16- 2x) * x\n\n=4x^(3) - 92x^(2) +480x \n\nTherfore  for largest volume: maximise V= 4x^(3) - 92x^(2) +480x \n\nTo maximise: first find values of x when the first derivative is equal 0:\n\nV ' = 12x^(2) -184x + 480 = 0\n\nx = 12 or 3 1/3\n\nNext verify which gives a maximum V value by either using first derivative test or second derivative test:\n\nWith first derivative test we choose values between intervals of x values for V'=0 and look at the trend:\n\nPositive values mean the function is increasing (/) and negative values means the function is deacreasing at that point (&#92;)\n\nx | 0 | 3 1/3 | 5 | 12 | 14                  \nV'| / .|... - .. | &#92;. |.. - ..| /\n\nTrend on either side of x= 3 1/3 :/&#92;\nTrend on either side of  x=12 : &#92;/\n\nTherefore max is at x= 3 1/3\n\nBy second derivative test :\nIf v"< 0 ie a negative number, V is a maximum\nIf v">0 ie positive number,V is a min\nIf V" = 0 it is a point of inflection\n\nV"= 24x-184\n\nx= 3 1/3,  V"= negative therefore max\nx= 12 ,    V"= positive therefore min\n\nHence the squares should be of lengths 3 1/3 inches to give the largest volume
1172136
try this and learn more about volume\nhttp://en.wikipedia.org/wiki/Volume
463030
Conversion Calculator here\nhttp://education.yahoo.com/reference/weights_and_measures/index?p=volume