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Math Help - And yet more calculus questions I need help on - Mostly with Maximums and Minimums...

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    And yet more calculus questions I need help on - Mostly with Maximums and Minimums...

    1.A company is informed by its market analyst, that if the price of widgets is $90-x, the company will sell 100+10x thousand widgets. At what price should the widget be set so the company maximizes income?


    2.Another company sells wrought iron umbrella trees. This company has fixed costs of $100,000/month. The cost in labor and material is $15 per umbrella sold per month. To sell x items, the price must be set at $50-0.7x, where x is in thousands of umbrellas sold per month. What price will maximize profit?

    3.A drum in the form of a circular cylinder and open at one of the circular ends, is to be made so as to contain one cubic yard. Find the dimensions of the drum (height h and base radius r ) which minimizes the amount of material going into the drum. The surface area of the drum includes the area of the cylindrical side and the circular disc at bottom.

    4.A rectangle with sides parallel to to the coordinate axes is inscribed in the ellipse

    16x^2 + 9y^2 = 144

    Find the dimensions of the rectangle of greatest area.
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    Quote Originally Posted by derekjonathon View Post
    1.A company is informed by its market analyst, that if the price of widgets is $90-x, the company will sell 100+10x thousand widgets. At what price should the widget be set so the company maximizes income?
    Income is "price time number sold" which, here, is (90-x)(100+ 10x). If you multiply that out, you get a quadratic function and can maximize it by completing the square.


    2.Another company sells wrought iron umbrella trees. This company has fixed costs of $100,000/month. The cost in labor and material is $15 per umbrella sold per month. To sell x items, the price must be set at $50-0.7x, where x is in thousands of umbrellas sold per month. What price will maximize profit?
    Again, the income is "price times number sold"= (40-0.7x)(x). But now must also subtract costs: 100000 and 15x. The profit is (40-0.7x)(x)- 1000000- 15x. Again that is a quadratic function that you can maximize by completing the square.

    3.A drum in the form of a circular cylinder and open at one of the circular ends, is to be made so as to contain one cubic yard. Find the dimensions of the drum (height h and base radius r ) which minimizes the amount of material going into the drum. The surface area of the drum includes the area of the cylindrical side and the circular disc at bottom.
    The area you want to minimize is 2\pi r h+ \pi r^2. The total volume is \pi r^2h= 1 so h= \frac{1}{\pi r^2}. Replace h in 2\pi h+ \pi r^2 by that to get a formula in r only. Minimize that.

    4.A rectangle with sides parallel to to the coordinate axes is inscribed in the ellipse

    16x^2 + 9y^2 = 144

    Find the dimensions of the rectangle of greatest area.
    Let (x,y) be the point where the corner or the rectangle lies on the ellipse. Then the width of the rectangle is 2x and the height is 2y so the area is (2x)(2y)= 4xy. Minimize that with the condition that 16x^2+ 9y^2= 144. You can reduce the area to a function of one variable by using y= 3\sqrt{144- 16x^2}
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