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Math Help - Optimization Problems (Calc)

  1. #1
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    Optimization Problems (Calc)

    Back again! For some reason, these remind me of related rates =P

    1) What is the smallest perimeter and dimensions of a rectangle whose area is 16 inches squared?
    (I know the answer is 16 and 4x4, I just don't know how to do it)

    Also, what is "Snell's Law"? It seems like we're using it in many problems, but because my teacher doesn't actually teach anything, I don't understand it

    Thanks!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Super Mallow View Post
    Back again! For some reason, these remind me of related rates =P

    1) What is the smallest perimeter and dimensions of a rectangle whose area is 16 inches squared?
    (I know the answer is 16 and 4x4, I just don't know how to do it)
    are you in calc, or precalc? there are different ways to do it depending on which class you're in

    Also, what is "Snell's Law"? It seems like we're using it in many problems, but because my teacher doesn't actually teach anything, I don't understand it

    Thanks!
    Snell's law is something used in Physics, specifically, optics.

    it states that as light moves from one medium to another, the following relationship holds:

    \frac {\sin \theta_1}{\sin \theta_2} = \frac {v_1}{v_2} = \frac {n_2}{n_1}

    where:
    \theta_1 is the angle of incidence
    \theta_2 is the angle of refraction
    v_1 is the speed of light in the first medium
    v_2 is the speed of light in the second medium
    n_1 is the refractive index of the first medium
    n_2 is the refractive index of the second medium

    see here for more info
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  3. #3
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    I'm in Calc

    I ended up getting this problem. I am stuck on another though

    4) A rectangle has its base on the X-Axis and it's upper two vertices on the parabola 12-xSquared. What is the largest area the rectangle can have, and what are the dimensions?

    Now that, I have NO idea how to solve...I just don't know the process
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Super Mallow View Post
    Back again! For some reason, these remind me of related rates =P

    1) What is the smallest perimeter and dimensions of a rectangle whose area is 16 inches squared?
    (I know the answer is 16 and 4x4, I just don't know how to do it)
    Let x and y be the length and width of the rectangle respectively

    the area is given by: A = xy

    we are told xy = 16 ............(1)

    we wish to minimize the perimeter, P, which is given by:

    P = 2x + 2y ..................(2)

    from (1), y = \frac {16}x

    so, P = 2x + \frac {32}x

    now find P' and set it equal to zero. solve for x and then for y and you can answer the question
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Super Mallow View Post
    I'm in Calc

    I ended up getting this problem. I am stuck on another though

    4) A rectangle has its base on the X-Axis and it's upper two vertices on the parabola 12-xSquared. What is the largest area the rectangle can have, and what are the dimensions?

    Now that, I have NO idea how to solve...I just don't know the process
    Draw a diagram. you will notice that the base of the rectangle is given by 2x and the height is given by y = 12 - x^2

    now, A = \mbox{length} \times \mbox{width} = 2x \left( 12 - x^2 \right)

    Now, again, find A', set it equal to zero and solve for x, then solve for y, and you will be able to answer the question
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