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DanielaJohana

  • one year ago

What is the maximum volume of a box that can be made from an 11 inch by 18 inch piece of paper? Show your work Will fan and medal

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  1. campbell_st
    • one year ago
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    let x he the height of the box... length = 18 - 2x width = 11 - 2x |dw:1433120436164:dw| the volume is \[V = x (18-2x)(11-2x)\] multiply it out

  2. campbell_st
    • one year ago
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    when you get an equation, then find the 1st derivative. let it equal zero and solve for x

  3. DanielaJohana
    • one year ago
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    \[V=x(18-2x)(11-2x) \] \[V=(18x-2x^2)(11x-2x^2)\] \[V=198x^2-36x^3-22x^3+4x^4\] \[V=198x^2-58x^3+4x^4\]

  4. campbell_st
    • one year ago
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    well you may want to check, as I think the equation is a cubic... V = x(198 - 58x + 4x^2) then distribute this V = 198x - 58x^2 + 4x^3 now find the 1st derivative

  5. DanielaJohana
    • one year ago
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    yes, it is cubic- and would it be 198-116x+12x^2?

  6. campbell_st
    • one year ago
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    now solve for x

  7. DanielaJohana
    • one year ago
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    x= 2.214 and 7.453 Now what do I do?

  8. campbell_st
    • one year ago
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    well you can't use 7.453... as the width is 11-2x is x is 7.453 the width is -4....which is impossible... so it must be the smaller choice...is probabbly the solution... just check by substituting

  9. campbell_st
    • one year ago
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    hope that makes sense... the alternative is to find the 2nd derivative... and test the solutions in the 2nd derivative to determine the max value

  10. DanielaJohana
    • one year ago
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    Yeah, once I plug it in it gives me 197.48 as the answer.. and that's the maximum volume right?

  11. campbell_st
    • one year ago
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    I haven't done the caclulations... you can check by graphing thecubic equation

  12. DanielaJohana
    • one year ago
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    Okay, I see how this worked out, thank you

  13. campbell_st
    • one year ago
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    I just graphed it and the smaller value gives the max volume

  14. campbell_st
    • one year ago
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    the max is (2.2, 197.5) hope it helped

  15. DanielaJohana
    • one year ago
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    yes, it did thanks

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