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Math Help - Help with Optimization Problems

  1. #1
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    Exclamation Help with Optimization Problems

    After the new semester started we ended up going back over optimization and I was gone sick for the lecture. I've been trying to figure these out for what seems like forever and I think I have lost all knowledge in this area. xD Help or a solution would be so appreciated.

    1. Your iron works has contracted to design and build a 500ft^3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, you job is to find dimensions for the base and height that will make the tank weigh as little as possible. What dimensions do you tell the shop to use?

    2. A 1125ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy. If the total cost is c=5(x^2 + 4xy) + 10xy what values of x and y will minimize it?

    3. Two sides of a triangle have lengths a and b, and the angle between them is ө. What value of ө will maximize the triangle's area? [hint: A = (1/2) ab sin ө.]

    If you can help with any of these I would be so thankful for I have to go back to highschool tomorrow and would at least like to have something to work with when I ask my Calc teacher for help. Thank you!
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  2. #2
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    Quote Originally Posted by Crowdia View Post
    1. Your iron works has contracted to design and build a 500ft^3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, you job is to find dimensions for the base and height that will make the tank weigh as little as possible. What dimensions do you tell the shop to use?
    Hi

    Let V_0 = 500 ft^3
    Let a be the size of the square-base
    Let h be the height of the tank

    Then a^2 h = V_0

    The tank being homogeneously made in steel, its weigh W is proportional to the volume of the 5 sheets that are necessary to build it.
    All the sheets having the same thickness, the weigh of the tank is proportional to the total surface S of the 5 sheets.

    W = \alpha S = \alpha (a^2 + 4 ah)

    W = \alpha \: \left(a^2 + 4 a \frac{V_0}{a^2}\right) = \alpha\: \left(a^2 + 4 \frac{V_0}{a}\right)

    Now you just have to differentiate W with respect to a to find the value of a that minimizes the weight.
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  3. #3
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    Hello, Crowdia!

    3. Two sides of a triangle have lengths a and b, and the angle between them is \theta.

    What value of \theta will maximize the triangle's area?

    Hint: . A \:=\:\tfrac{1}{2}ab\sin\theta
    Differentiate and equate to zero . . .

    . . \frac{dA}{d\theta} \:=\: \tfrac{1}{2}ab\cos\theta \:=\:0 \quad\Rightarrow\quad \cos\theta \:=\:0 \quad\Rightarrow\quad \theta \:=\:\frac{\pi}{2} \:=\:90^o


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    With a little Thought, you can "eyeball" the solution.
    Code:
                *
               /: . 
           a  / :   . 
             /  :h    .
            /   :       . 
           / θ  :         .
          * - - + - - - - - *
          :  - - - b - - -  :

    Since A \:=\:\tfrac{1}{2}bh, the area is a maximum when h is at its maximum.

    And this happens when h = a:\;\theta \,=\,90^o.

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  4. #4
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    Thank You!

    Thank you guys soooooo much! I can't believe how much help I was able to get in such a short amount of time. You guys are amazing!
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