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6. A parabaloid with height h and radius r has volume 1/2 pieR^2H. Consider a parabaloid-shaped water
tank with height of 7m and radius of 4m. If V is the volme of water in the tank and h is the depth
of water,
find the relationship between dV/dt and dhdt.
How is the volume changing when h = 3m and
dh/dt = 0.2m/s?

  • calculus -

    dV/dt = surface area * dh/dt (geometry , no calculus really required, draw picture)
    dV/dt = pi r^2 dh/dt

  • calculus -

    consider a cross-section of the paraboloid placed with vertex at the origin an opening upwards
    then its equation would be
    y = ax^2
    when x=4, y = 7
    y = (7/16)x^2
    h = (7/16)r^2 , using r and h
    r^2 = 16h/7

    V = (1/2)πr^2h
    = (1/2)π(16h/7)h
    = (8π/7) h^2
    dV/dt = 16π/7 h dh/dt ----> the relationship between dV/dt and dh/dt

    Whe h=3 and dh/dt = .2

    dV/dt = (16π/7)(3)(.2) = ......

    I will let you do the button pushing.

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