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Math Help - Calc IV: Functions of Several Variables (Chain Rule)

  1. #1
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    Calc IV: Functions of Several Variables (Chain Rule)

    Hello,
    I am having a hard time solving this problem. I have no clue. None whatsover. Thanks in advance for your help.

    Volume & Surface Area
    The radius of a right circular cone is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What are the rates of change of the volume and surface area when radius is 12 inches and height is 36 inches?
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  2. #2
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    Hello,

    V=(\pi r^2)\frac h3=\frac{\pi}{3} r^2 h

    Where r is the radius and h the height.

    You're looking for \frac{dV}{dt}

    By using the multiplication rule, you have :

    \frac{dV}{dt}=\frac{\pi}{3} \left( \frac{dh}{dt} r^2+2h \frac{dr}{dt} r \right)

    From the text, we know that \frac{dh}{dt}=-4, \ \frac{dr}{dt}=6, \ r=12, \ h=36

    So it should give you :

    \frac{dV}{dt}=\frac{\pi}{3} (-4*144+2*36*6*12)=\dots
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    I believe your formula for the cone is \frac{1}{3}pi r^2h for the cone. Would you just not take the derivative as you insert your variables. I didn't know there was a Cal IV I thought they only went to III (just an aside to myself )
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    Yeah, listen to Moo, as he understands volume formula much better than myself
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  5. #5
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    For the area, it's the sum of the area of the base and the lateral area.

    The area of the base is given by \pi r^2

    The lateral area is given by \pi r \sqrt{h^2+r^2} (I can try to explain it, but later...)

    Hence A=\pi r^2+\pi r \sqrt{h^2+r^2}

    Once again, you're looking for \frac{dA}{dt}

    Can you do it ?
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    hello,
    I am not so sure about the second part (lateral area) and the dA/dt what is the dA/dt? can you explain in details please because I am little confused on the second part Thanks!
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    Moo
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    It means "the derivative of A in respect with t".
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    How do you have take the derivative , I forgot whether I use the chain rule or
    other rules, if so how do I do it?
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    Quote Originally Posted by googoogaga View Post
    How do you have take the derivative , I forgot whether I use the chain rule or
    other rules, if so how do I do it?
    Are you needing reminders on how to take a derivative in general or how to take a partial derivative? With partial derivatives you want to take the derivative in the respect to the variable in question. An example would be like

    x^2+y^2+3 in respect of x would be 2x

    Check out some rules online if your still a bit confused on it
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  10. #10
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    I am so sorry for the late thanks I just got so caught up with school work, you know it's time for exams. Thanks again! You clarified it!
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  11. #11
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    You clarified it!
    You know, this is the most important thing... Sincerely
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