Caclulus

Let x and y be to positive numbers whose product is 500:
(a) Find the maximum sum of x and y
(b) Find the minimum sum of x and y:
2. A cylindrical container can hold a volume of 1 liter. Find the dimensions
of the container that minimizes the surface area.
3. What are the dimensions of the rectangle with largest area that can be
drawn inside a circle of radius R?
4. Find the maximal area of a right triangle with hypotenuse of length L:

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  1. #1 I assume that x,y,z are integers, or there is no maximum sum. The minimum sum would be 3∛500 ≈ 23.811

    500 = 2^2 5^3 so its factors are
    1,2,250
    1,4,125
    1,5,100
    1,10,50
    1,20,25
    2,2,125
    2,5,50
    2,10,25
    5,5,20
    5,10,10
    5,25,4
    Now pick the largest and smallest sums.

    #2 v = πr^2h = 1
    so, h = 1/(πr^2)

    The surface area is

    a = 2πr^2 + 2πrh
    = 2πr^2 + 2πr/(πr^2)
    = 2πr^2 + 2/r
    da/dr = 4πr - 2/r^2
    da/dr=0 when r=1/∛(2π)
    so, h=∛(4/π)

    #3 The diagonal of the rectangle is the diameter of the circle. So,

    x^2+y^2 = 4r^2
    a = xy = x√(4r^2-x^2)
    da/dx = 2(2r^2-x^2)/√(4r^2-x^2)
    da/dx=0 when x = r√2
    So, as expected, the rectangle is a square with diagonal 2r

    #4 as above, it will be half of a square, with legs L/√2.
    It's really the same problem, because a right triangle can be inscribed in a circle. Its hypotenuse is the diameter of the circle.

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