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Thread: Polynomial Functions

  1. #1 Polynomial Functions 
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    K i got this application question the other day in my introduction to calculus class and i was wondering if anyone could lend me a hand.

    Katie is building a wooden rectangular storage box. The box will have an open top and a volume of 1.5m cubed. For design purposes, Katie would like the length of its base to be triple its width. Thick wood for the base costs $8/m squared and thinner wood for the sides costs $5/m squared.
    a) Express the cost of the wood as a finction of the width of the base.
    b) Find all possible dimensions if Katie spends $44 for the wood.
    c) what dimensions would you recommend? Why?

    Now, i think if someone could help me with a) then i probably would be able to get the rest. I have figured out the base of the box is 3w + w, where w represents the the width size, but given that i do not know how to get the height of the box. I believe if i can determine the w variable then i could find the height by manipulating the formula for the volume of a box (V= lwh). anyways any help would be greatly appreciated..


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  3. #2  
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    Hi, I'll help you do the first part but I must warn you that I cannot guarantee that I am correct. Here goes.

    Katie is building a wooden rectangular storage box. The box will have an open top and a volume of 1.5m cubed. For design purposes, Katie would like the length of its base to be triple its width. Thick wood for the base costs $8/m squared and thinner wood for the sides costs $5/m squared.
    a) Express the cost of the wood as a finction of the width of the base.
    Let's break this down: Katie is building a wooden rectangular storage box.
    What's the volume of this box? 1.5m cubed, but let's keep it general for now. We'll call it V. V=length*width*height
    Let's call length L, width w, height h.
    V=Lwh.......(1) {V=1.5m^3}

    Katie would like the length of its base to be triple its width. So how do you express that in general terms?
    L=3w.........(2)

    The box will have an open top; thick wood for the base costs $8/m squared and thinner wood for the sides costs $5/m squared.

    Let's call the cost of base wood B {B=8$/m^2}, and the cost of side wood S {S=5$/m^2}.

    Now you should be able to see that the amount of base wood needed is equal to the area of the base (Lw), and the amount of side wood is equal to the combined area of the two walls along the length 2(Lh), and the two walls along the width 2(wh), i.e. 2(Lh+wh).

    So what is the cost C of the wood? It's the cost the base wood plus the cost of the side wood.
    C=BLw+2S(Lh+wh).......(3)

    Now to answer your question you need to try and get C in terms of just w, B, S, and V. Then you can substitute in the known values of B, S, and V to obtain an expression for C in terms of just w (or you might want to plug your known values in right at the beginning). You'll need to do a little rearranging but really it's not that hard, I'll leave you the satisfaction of finding an answer and we can compare later if you'd like.

    Cheers


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  4. #3  
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    thanks a lot, but the problem the height is not given so some how i have to find it. Also, when i said the volume was 1.5 to the cubed, i didn't mean to the exponent and i am sorry for the confusion...the same thing for the cost of the wood. anyways i'll try more to look at what you have given be but i still dont know how to do this
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  5. #4  
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    Don't worry about the powers, that's just dimensions of length (you'll learn all about these if you do physics). You know, like areas are m^2, volumes are m^3; that's the same as saying areas are metres squared and volumes are metres cubed.

    Okay I'll simplify it for you, let's re-write (1), (2), and (3) and put in the values for V, B, and S.

    1.5=Lwh....................(1)
    L=3w.........................(2)
    C=8Lw+10(Lh+wh).....(3)

    Rearrange (1): h=1.5/Lw, then combine with (2):

    h=1.5/3w^2..........(4)

    Now sub (2) and (4) into (3). It should be pretty easy.
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  6. #5  
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    wow thanks a lot...it looks like it will work, but i'll have to check later cuz i'm gonna out with my gf...once again thanks
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