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Math Help - Differentiation Help?

  1. #1
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    Differentiation Help?

    A globe with a diameter of 30 cm is to be painted with a uniform layer of paint which is 4 mm thick. Use the method of differentiation to find the approximate volume of paint needed.

    Help anyone?
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  2. #2
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    Quote Originally Posted by xje4bv View Post
    A globe with a diameter of 30 cm is to be painted with a uniform layer of paint which is 4 mm thick. Use the method of differentiation to find the approximate volume of paint needed.

    Help anyone?
    Globe, sphere:
    Volume, V= (4/3)pi(r^3)
    Differentiate both sides,
    dV = (4/3)pi[3r^2 *dr]
    dV = 4pi(r^2)(dr)

    When r = 30/2 = 15cm, and dr = 4mm = 0.4cm,
    dV = 4pi(10^2)(0.4)
    dV = 160pi cu.cm = 502.65 cc ----answer.
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  3. #3
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    Quote Originally Posted by ticbol View Post
    Globe, sphere:
    Volume, V= (4/3)pi(r^3)
    Differentiate both sides,
    dV = (4/3)pi[3r^2 *dr]
    dV = 4pi(r^2)(dr)

    When r = 30/2 = 15cm, and dr = 4mm = 0.4cm,
    dV = 4pi(10^2)(0.4)
    dV = 160pi cu.cm = 502.65 cc ----answer.
    Urm, thanks for replying, but the answer is 1440 pi cm cube. Anyone know how and why?
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  4. #4
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    Quote Originally Posted by xje4bv View Post
    Urm, thanks for replying, but the answer is 1440 pi cm cube. Anyone know how and why?
    Ooops, I used r=10cm instead of the r=15cm
    So, correction,
    dV = 4pi(15^2)(0.4) = 360pi cu.cm.

    Not the 160pi cu.cm I wrote before.

    But the answer should be 1440pi cu.cm?

    Let us check by geometry.
    >>>sphere with no paint, r = 15cm
    V1 = (4/3)pi(15^3) = 4500pi cc.

    >>>sphere painted 4mm thick, r = 15.4 cm
    V2 = (4/3)pi(15.4^3) = 4869.7 cc

    So, volume of paint = V2 -V1 = 4869.7pi -4500pi = 369.7pi

    That is much closer to the 360pi cc than to the 1440pi cc.
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  5. #5
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    Quote Originally Posted by ticbol View Post
    Ooops, I used r=10cm instead of the r=15cm
    So, correction,
    dV = 4pi(15^2)(0.4) = 360pi cu.cm.

    Not the 160pi cu.cm I wrote before.

    But the answer should be 1440pi cu.cm?

    Let us check by geometry.
    >>>sphere with no paint, r = 15cm
    V1 = (4/3)pi(15^3) = 4500pi cc.

    >>>sphere painted 4mm thick, r = 15.4 cm
    V2 = (4/3)pi(15.4^3) = 4869.7 cc

    So, volume of paint = V2 -V1 = 4869.7pi -4500pi = 369.7pi

    That is much closer to the 360pi cc than to the 1440pi cc.
    According to my textbook, the answer is 1440 pi. Anyway, my teacher said that maybe there are mistakes in the textbook, but being an obstinate person, I decided to consult you guys. Well, if anyone is able to find an answer as to why the answer stated is 1440 pi cm cube, please do not hesitate to post your solution. Thanks anyway to ticbol for your fast reply.
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  6. #6
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    Is this right? I drew a diagram and tried separating the paint into a cuboid by using 0.4 cm as the height, 30 pi (circumference) as the length, and dA over dr, 120 pi (surface area) as the depth. My answer is 1440 pi^2 cm cube which looks a lot like the answer, but I'm sure of otherwise. Help anyone?
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