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Math Help - Growing tree

  1. #1
    a.a
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    Growing tree

    The trunk of a tree is approximately cylindrical in shape and has a diameter of 1 m when the height is 15m. If the radius is inreasing at 3mm/a and the height is increasing at 0.4 m/a, find the rate of increase of the volume of the trunk.

    using related triangles i got r = h/30 and h=30r
    and using the volume of a cylinder i got V = ( pi h^3)/900
    then i said that dV/dh = (pi h^2)/ 300
    also dV/dt= dV/dh times dh/dt and i got dV/dt = (pi h^2)/ 750

    then just to check i used subed in h = 30r in the volume formula and did all the same steps and got a different answer... so i kno im doing something wrong.. :S

    can u please help me out here? i feel realli stupid
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  2. #2
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    Hello, a.a!

    The trunk of a tree is approximately cylindrical in shape
    and has a diameter of 1 m when the height is 15 m.
    If the radius is increasing at 3mm/year (0.003 m/year)
    and the height is increasing at 0.4 m/year,
    find the rate of increase of the volume of the trunk. . . . . when?

    The volume of a cylinder is: . V \;=\;\pi r^2h

    At t = 0\:\;\;r = 0.5,\:h = 15

    After t years: . \begin{array}{ccc}r &=&0.5 + 0.003t \\ h &=& 15 + 0.4t\end{array}

    Then: . V \;=\;\pi(0.5+0.003t)^2(15 + 0.4t) \;=\;\pi(0.000036t^3 + 0.00255t^2 + 0.55t + 3.75)


    Therefore: . \frac{dV}{dt} \;=\;\pi(0.000108t^2 + 0.0051t + 0.55) \;\text{ m}^3\text{/year}
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  3. #3
    a.a
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    Quote Originally Posted by Soroban View Post
    Hello, a.a!


    The volume of a cylinder is: . V \;=\;\pi r^2h

    At t = 0\:\;\;r = 0.5,\:h = 15

    After t years: . \begin{array}{ccc}r &=&0.5 + 0.003t \\ h &=& 15 + 0.4t\end{array}

    Then: . V \;=\;\pi(0.5+0.003t)^2(15 + 0.4t) \;=\;\pi(0.000036t^3 + 0.00255t^2 + 0.55t + 3.75)


    Therefore: . \frac{dV}{dt} \;=\;\pi(0.000108t^2 + 0.0051t + 0.55) \;\text{ m}^3\text{/year}
    can we just differentiate V = pi r^2 w.r.t t and use product rule?
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