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Maxima word problem - Not sure about my answer.

  1. Sep 15, 2012 #1
    Maxima word problem -- Not sure about my answer.

    This was in an assignment handed out after our introduction to extrema's.
    A builder wants to build a small building consisting of 4 walls and a roof (all rectangular). The cost of the bricks for the walls will be $25/square metre, while the roof materials will cost $55/square metre. The building must have a square base, but otherwise may have any dimensions the builder wishes. If the builder’s budget is $1650 then what dimensions would make the build have the largest possible volume?

    The attempt at a solution
    Premise: A cube will maximize the volume for surface area in this solution.

    Roof (A) = L2
    Walls (B) = 4 × (L2)

    $1650 = $55A + $25(4B)
    $1650 = $55A + $100B

    $1650 = $55L2 + $100L2
    $1650 = $155L2
    $1650 / $155 = L2

    L = √($1650 / $155)

    L≈3.262692338m
    L≈3.263m (3.d.p)

    For a calculus problem, I am scared that I have not had to use calculus. :/
    There was another approaches before I used this method..
    Such as..

    ================================================
    Roof (A) = L2
    Walls (B) = 4 × (HL) <------------Where h is an variable height.

    1650 = A + 4B
    1650 = L2 + 4HL
    1650 = 55L2 + 100HL

    55L2 + 100HL -1650 = 0

    But, I couldn't think beyond this.. :/

    Also, using 1650 = 55L2 + 100HL,
    I managed to arbitrarily plug and chug a result of L=3m H=3.85m
    ================================================
    EDIT:

    I've established that the Volume would be L2H

    V = L2H
    A = L2
    B = HL
     
    Last edited: Sep 16, 2012
  2. jcsd
  3. Sep 16, 2012 #2

    Simon Bridge

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    Re: Maxima word problem -- Not sure about my answer.

    Have you tried finding an expression for the volume in terms of H or L alone?
     
  4. Sep 16, 2012 #3
    Re: Maxima word problem -- Not sure about my answer.

    In which way?

    I could make V= 55L2H
    {Area of the base × height.}

    I haven't tried using my premise that the most efficient {volume:surface area} is a cube, with the implication of V= L3

    I'll give that a shot.

    Otherwise, I am truly stumped.
     
    Last edited: Sep 16, 2012
  5. Sep 16, 2012 #4

    Simon Bridge

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    Re: Maxima word problem -- Not sure about my answer.

    No - I mean you have two simultaneous equations and three unknowns ... why not eliminate one of the unknowns in the equation that starts V= ?

    For instance ... from 1650=55L2+100HL ... you can solve it for H, then substitute into V=HL2 ... which gives you V in terms of L. Now you can find the maxima.

    You could assume that the shape is a cube - but you still have to prove that a cube is the right shape.
     
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