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Math Help - Rate of Change - Volume and Surface Area of Cylinder

  1. #1
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    Rate of Change - Volume and Surface Area of Cylinder

    We have a special type of cylinder whose height, h, is always twice its radius r. What is the rate of change of its volume V with respect to r in terms of the total surface area, A?

    I figured V= (pi)(r^2)(h) and since h=2r , V = (pi)(r^2)(2r)
    Surface Area = 2(pi)(r^2)+(pi)(r)(h) = 2(pi)(r^2)+(pi)(2r^2)

    I'm really not sure what to do after that though, thanks.

    - Adrian
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  2. #2
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    Quote Originally Posted by Adrian View Post
    We have a special type of cylinder whose height, h, is always twice its radius r. What is the rate of change of its volume V with respect to r in terms of the total surface area, A?

    I figured V= (pi)(r^2)(h) and since h=2r , V = (pi)(r^2)(2r)
    Surface Area = 2(pi)(r^2)+(pi)(r)(h) = 2(pi)(r^2)+(pi)(2r^2)

    I'm really not sure what to do after that though, thanks.

    - Adrian
    I'm afraid your total surface area, A, is lacking.

    The problems asks us to find dV/dr in terms of A.

    V = pi(r^2)h = pi(r^2)(2r) = 2pi(r^3)

    A = 2[pi(r^2)] +2pi(r)*h
    A = 2pi(r^2) +2pi(r)(2r)
    A = 2pi(r^2) +4pi(r^2)
    A = 6pi(r^2)

    So,
    V = 2pi(r^3)
    = [pi(r^2)](2r)
    = [A/6](2r)
    = (1/3)A*r

    Differentiate that with respect to r,
    dV/dr = (1/3)A ----------------------------answer.
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  3. #3
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    Thanks

    That was very helpful, and generally the path I was trying to take, I think my problem was the surface area, which you helped with a lot, thanks again.
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