Calculus

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"A rectangle is inscribed in a semicircle of radius 2 cm. Find the largest area of such a rectangle".

There is a diagram, but I think the question makes it clear enough what is going on. I'm having problems finding a relationship that I can work with.

  • Calculus -

    draw a radius from the centre to a vertex of the rectangle
    Let the base of the rectangle be 2x, making the triangle base x, and let the height of the rectangle by y

    Area of rectangle = 2xy
    but x^2 + y^2 = 4
    y = √(4 - x^2) or (4-x^2)^(1/2)

    A = 2x(4-x^2)^(1/2)

    take the derivative using the product rule,
    set it equal to zero and solve for x

  • Calculus -

    Thank you! Never occurred to me to position the radius like that.

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