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anonymous

  • 5 years ago

If 1200cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box

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  1. amistre64
    • 5 years ago
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    the volume of a box is given by: V = lwh length times width times height right? the surface area of this box is equal to 4 sides + a bottom.. As = lw + 2lh + 2wh

  2. amistre64
    • 5 years ago
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    we know 1200 is the total surface area so.... 1200 = lw +2wh + 2lh

  3. amistre64
    • 5 years ago
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    if we can get the value of l,w,and h into single terms wed be doing good :)

  4. radar
    • 5 years ago
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    labis, since you were told that the base is a square, this can be simplified. Let x equal the dimension of the base. Now the volume is simply the area of the base times the heigth or \[V=x ^{2}h\]

  5. radar
    • 5 years ago
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    The surface area is now simplified as it is equal to the area of the base plus the area of the 4 sides:\[A=x ^{2}+4xh\] This is given to equal 1200 cm. \[x ^{2}+4xh=1200 cmsq\]

  6. radar
    • 5 years ago
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    From this equation express h in terms of x. This can be done as follows: Subtract x^2 from both sides giving. 4xh=1200-x^2 Now divide through (both sides of equation) by 4x.\[h=(1200-x ^{2})/4x\]

  7. radar
    • 5 years ago
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    Using standard rules for simplifying fractions, adding, subtracting etc. This becomes: h= 300/x - x/4 Now plug this value of h into the volume equation above. You then get: \[V=x ^{2}(300/x - x/4)\]

  8. radar
    • 5 years ago
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    This reduces to: \[V =300x-1/4x ^{3}\] Now differentiate and set to zero\[300-3/4x ^{2}=0\] Subtract 300 from both sides \[-3/4x ^{2}=-300\] Multiply both sides by 14/3 getting:\[x ^{2}=400\]

  9. radar
    • 5 years ago
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    Solve for x getting 20 cm. You now can solve for h getting 10 cm V is the sum of x squared times h or V=4000 cubic cm.

  10. radar
    • 5 years ago
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    I hoped you follow with understanding.

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