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Math Help - quadratic function application

  1. #1
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    quadratic function application

    I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

    1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost $8 per linear foot, while the wooden fence costs $2 per linear foot.

    a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

    b) If you have only $500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?


    And the second:

    2) A box is made from a 16 inch by 28 inch piece of cardboard by cutting equal squares from each corner and folding up the sides.

    a) Write a function expressing the volume of the box as a function of x.
    b) Determine the domain of this function
    c) Determine the size of the square that will produce a maximum volume for the box. Give a complete explanation of your solution.


    I understand it's a lot, but I would appreciate any sort of help. I'm really struggling with this. Thanks.

    - Paul
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  2. #2
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by paul View Post
    I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

    1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost $8 per linear foot, while the wooden fence costs $2 per linear foot.

    a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

    Let's say the brick wall is one of the "length" sides.

    The equation for perimeter is: P = l + l + w+ w

    And since the perimeter is also the amount of fence, the cost of the fence is: C = $2l + $8l + $2w + $2w

    b) If you have only $500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?
    Well then, you have the equation: 500 = $2l + $8l + $2w + $2w

    Substitute: 500 = 2l + 8l + 2w + 2w

    Add: 500 = 10l + 4w

    Subtract 10l from both sides: 500 - 10l = 4w

    Divide both sides by 4: 125 - 2.5l = w

    Now you want to find the largest area, so the equation for Area is: A = lw

    Substitute: A = l(125-2.5l)

    So: A = 125l - 2.5l^2

    Let's say that A = y and l = x: y = 125x - 2.5x^2

    Now graph that and find what value of x gives the largest area.
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  3. #3
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    Quote Originally Posted by Quick View Post

    Now graph that and find what value of x gives the largest area.
    Let me teach you a rule about quadradic functions that might help.

    If a>0
    Then,
    y=ax^2+bx+c
    Is a parabola that opens up.
    Thus it has a minimum value.
    Which is located at,
    x=-b/(2a)
    (Actually that is where derivative is zero).

    If a<0
    Then,
    y=ax^2+bx+c
    Is a parabola that opens down.
    Thus it has a maximum value.
    Which is located at,
    [tex]x=-b/(2a)

    Thus, to find a max/min for "a" non-zero
    You need to compute,
    -b/(2a)
    And determine what the sign of 'a' is.
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  4. #4
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    Quote Originally Posted by paul View Post
    I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

    1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost $8 per linear foot, while the wooden fence costs $2 per linear foot.

    a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

    b) If you have only $500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?


    And the second:

    2) A box is made from a 16 inch by 28 inch piece of cardboard by cutting equal squares from each corner and folding up the sides.

    a) Write a function expressing the volume of the box as a function of x.
    b) Determine the domain of this function
    c) Determine the size of the square that will produce a maximum volume for the box. Give a complete explanation of your solution.


    I understand it's a lot, but I would appreciate any sort of help. I'm really struggling with this. Thanks.

    - Paul
    Hello, Paul,

    you have to cut off 4 equal squares which have the side x. Thus the greatest square to cut of has a side length of 8".

    The base area of the box is:
    A=(16-2x)*(28-2x)=4x^2-88x+448, 0 ≤ x ≤ 8

    the volume of the box is:

    V=x * A=4x^3-88x^2+448x

    The volume will become an extreme value (Minimum or maximum) if the first derivative equals zero:

    V'(x)=12x^2-176x+448. Now calculate:

    0=12x^2-176x+448. Use the formula to solve a quadratic equation. I've got the results:

    x=22/3-2/3*sqrt(37) ≈ 3.278...[/tex]
    or
    x=22/3+2/3*sqrt(37) ≈ 11.388...[/tex]

    The 2nd value doesn't belong to the domain of the volume function.

    Plug in the 1st value into the volume function and you'll get the greatest Volume: V = 663.851 cubic inches.

    EB

    PS.: I've got some troubles to use Latex, so I send you this text as plain text. I hope you can use it.
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  5. #5
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    Hi,

    I've attached a sketch of your problem.

    EB
    Attached Thumbnails Attached Thumbnails quadratic function application-schachtel_ohne_deckel.gif  
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  6. #6
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by earboth View Post
    Hi,

    I've attached a sketch of your problem.

    EB
    Here's a good sketch of the problem
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