calculus

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A piece of elastic is attached to two nails on a flat board, with a button attached to the midpoint of the elastic. The ails are 5 cm apart. You stretch the elastic by pulling the button along the board i the direction that is perpendicular to the line between the nails.
A. Find the equation that relates the total length of the elastic x to the distance y that the button has moved.
B. You pull down the button at a constant 3cm/sec. Find the rate at which the length of the elastic is increasing when it is 12cm long.

  • calculus -

    each side of the elastic is the hypotenuse of a triangle with legs 5/2 and y, so

    x = 2√(2.5^2 + y^2)

    dx/dt = 2y/√(2.5^2+y^2) dy/dt
    when x=12, y=5.45, so
    dx/dt = 2(5.45)/6 (-3) = -1/5.45 = -0.18 cm/s

  • calculus -

    A.
    x = 2√[ y^2 + (2.5)^2 ] cm (using Pythagorus theorem)

    B.
    x^2 = 4y^2 + 25
    => 2x dx/dt = 8y dy/dt
    => dx/dt = 4 (y/x) dy/dt
    When x = 12 cm, y = (1/2)√[ 144 - 25 ] = 2.727 cm
    => dx/dt = 4 (2.727/12) x 3 cm/s
    = 2.727 cm/s.

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