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Thread: Composite function problem

  1. #1
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    Composite function problem

    Here is the problem I'm trying to solve:

    The volume of a conical pile of sand is increasing at a rate of 243(pi)ft^3/min, and the height of the pile always equals the radius r of the base. Express r as a function of time t (in minutes), assuming that r = 0 when t = 0.

    I'm not even sure how to set this one up! Please help!
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  2. #2
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    Hello, Tubaman!

    $\displaystyle \text{The volume of a conical pile of sand is increasing at }243\pi\text{ft}^3\!/\text{min}$
    $\displaystyle \text{The height }h\text{ of the pile always equals the radius }r\text{ of the base.}$
    $\displaystyle \text}Express }r\text{ as a function of time }t\text{, assuming that }r\,\!=\,\!0 \text{ when }t\,\!=\,\!0.$

    The volume of a cone is: .$\displaystyle V \;=\;\frac{\pi}{3}r^2h$

    Since $\displaystyle r = h$, we have: .$\displaystyle V \;=\;\frac{\pi}{3}r^3$

    Differentiate with respect to time: .$\displaystyle \dfrac{dV}{dt} \;=\;\pi r^2\dfrac{dr}{dt} $


    We are told that: $\displaystyle \dfrac{dV}{dt} \,=\,243\pi\text{ ft}^3\!/\text{min} $

    Hence: .$\displaystyle \pi r^2\dfrac{dr}{dt} \;=\;243\pi \quad\Rightarrow\quad r^2\,dr \;=\;243\,dt $


    Integrate: .$\displaystyle \frac{1}{3}r^3 \;=\;243t + c \quad\Rightarrow\quad r^3 \;=\;729t + C$

    When $\displaystyle r=0,\:t=0\!:\;\;0^3 \:=\:729(0) + C \quad\Rightarrow\quad C \,=\,0$

    So we have: .$\displaystyle r^3 \:=\:729t$


    Therefore: .$\displaystyle r \;=\;9\sqrt[3]{t}$
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  3. #3
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    Non-calculus answer

    Soroban,

    Thanks much for your response. However, this problem was presented within a chapter regarding composite functions; that is, to solve the problem using "nested" functions. Is it possible to present a solution that utilizes this method, and that also employs no calculus?

    Thanks!
    Tubaman
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