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Math Help - [SOLVED] Rational Function: Application word problem

  1. #1
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    [SOLVED] Rational Function: Application word problem

    I need help understanding the process to combine terms to result in a specific function. I have a cylinder with the volume and surface area defined below:

    V=Pi(r^2)h
    S=2(Pi(r^2)+2(Pi)rh

    The book is asking: Assume that the volume of the cylinder is 1000 cm^3. Express the surface area as a function of the radius. After combining terms in your answer, show that the resulting function can be written:

    S(r) = (2(Pi)(r^2) + 2000)/r where (r>0)

    I understand that the problem is looking for S(r), where 2(Pi)(r^2)+2(Pi)rh is the formula for the surface area, but I don't understand how the book ends up with the resulting function.

    Thank you!
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  2. #2
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    Hello, Snowcrash!

    If you typed everything correctly, their answer is wrong . . .


    I have a cylinder with its volume and surface area: . \begin{array}{cccc}(1) & V &=&\pi r^2h \\ (2) &S &=&2\pi r^2 + 2\pi rh \end{array}

    Assume that the volume of the cylinder is 1000 cm³.
    Express the surface area as a function of the radius.
    Show that the function can be written: . S(r) \:= \:\frac{2\pi r^2 + 2000}{r} .
    . . . This is wrong!
    From (1), we have: . \pi r^2h \:=\:1000 \quad\Rightarrow\quad h \:=\:\frac{1000}{\pi r^2}

    Substitute into (2): . S \;=\;2\pi r^2 + 2\pi r\left(\frac{1000}{\pi r^2}\right)

    Therefore: . S \;=\;2\pi r^2 + \frac{2000}{r}

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  3. #3
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    Quote Originally Posted by Snowcrash View Post
    I need help understanding the process to combine terms to result in a specific function. I have a cylinder with the volume and surface area defined below:

    V=Pi(r^2)h
    S=2(Pi(r^2)+2(Pi)rh

    The book is asking: Assume that the volume of the cylinder is 1000 cm^3. Express the surface area as a function of the radius. After combining terms in your answer, show that the resulting function can be written:

    S(r) = (2(Pi)(r^2) + 2000)/r where (r>0)

    I understand that the problem is looking for S(r), where 2(Pi)(r^2)+2(Pi)rh is the formula for the surface area, but I don't understand how the book ends up with the resulting function.

    Thank you!
    You have V=1000
    and V=\pi r^2h

    hence, 1000=\pi r^2 h

    Can you continue now?



    EDIT: beaten by Soroban....he will be glad.
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    Thank you Soroban and Malay -

    I double-checked and the function that I did type the function incorrectly, in that I used a square instead of cube (2(Pi)(r^3):


    S(r) = (2(Pi)(r^3) + 2000)/r where (r>0)

    Would the cube come in because the volume is 1000cm^3??

    My first instinct would be to solve the volume formula for the radius because the problem is asking me to "express the surface area as a function of the radius" or s(r) instead of solving for the h value.

    Like r^2 = 1000/(Pi(h))

    But you are solving for h instead - how did you know to solve for h?
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  5. #5
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    Quote Originally Posted by Snowcrash View Post
    Thank you Soroban and Malay -

    I double-checked and the function that I did type the function incorrectly, in that I used a square instead of cube (2(Pi)(r^3):


    S(r) = (2(Pi)(r^3) + 2000)/r where (r>0)

    Would the cube come in because the volume is 1000cm^3??

    My first instinct would be to solve the volume formula for the radius because the problem is asking me to "express the surface area as a function of the radius" or s(r) instead of solving for the h value.

    Like r^2 = 1000/(Pi(h))

    But you are solving for h instead - how did you know to solve for h?
    Hi Snowcrash,

    You said you needed the Surface Area stated as a function of the radius.

    Solving for h in the Volume formula and substituting this into the Surface Area formula gave you the Surface Area as a function of the radius. You are not solving for r. You're just stating the function in terms of r.


    S(r)=2\pi r^2+\frac{2000}{r}

    \pi is a constant.
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  6. #6
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    Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

    <br />
S(r) \:= \:\frac{2\pi r^3 + 2000}{r}<br />

    Soroban said that the answer is incorrect - but I wrote it incorrectly with <br />
S(r) \:= \:\frac{2\pi r^2 + 2000}{r}<br />

    How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?
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  7. #7
    Senior Member Stroodle's Avatar
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    Quote Originally Posted by Snowcrash View Post
    Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

    <br />
S(r) \:= \:\frac{2\pi r^3 + 2000}{r}<br />

    Soroban said that the answer is incorrect - but I wrote it incorrectly with <br />
S(r) \:= \:\frac{2\pi r^2 + 2000}{r}<br />

    How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?
    \frac{2\pi r^3 + 2000}{r} =\frac{2\pi r^3}{r}+\frac{2000}{r}=2\pi r^2+\frac{2000}{r}
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  8. #8
    A riddle wrapped in an enigma
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    Quote Originally Posted by Snowcrash View Post
    Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

    <br />
S(r) \:= \:\frac{2\pi r^3 + 2000}{r}<br />

    Soroban said that the answer is incorrect - but I wrote it incorrectly with <br />
S(r) \:= \:\frac{2\pi r^2 + 2000}{r}<br />

    How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?
    Snowcrash,

    S=2\pi r^2+\frac{2000}{r} can be rewritten as

    S=\frac{2\pi r^3+2000}{r}

    It's just another way of expressing the same function. All we did here was put the right hand expression over one denominator.
    S=2\pi r^2+\frac{2000}{r} \implies S=\frac{2\pi r^3}{r}+\frac{2000}{r} \implies S=\frac{2\pi r^3+2000}{r}
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  9. #9
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    Thanks, I see it now. Thank you for your help everyone!
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