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Math Help - Solids of revolution, spring constants, and limits

  1. #1
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    Solids of revolution, spring constants, and limits

    These are some problems that I'm having trouble with from a review packet. I know the answers, but I don't know how to do them. I would greatly appreciate any help I could get, even if you only help on one or two. I included my work for some of them so you can tell me what I'm doing wrong.


    1.) Find lim t --> 0 of [ln(x-9)^8] / [2x - 20]
    I used L'hospital's rule a few times and got 0, but the answer is 4. I see 8/2 in the last part, but I thought the ln in the numerator approaches 0...?








    2.) The region y = 2x - x^2 and y = 0 is rotated around the x-axis. Find the value of a if the resulting volume is 44π / 3.

    The answer is 3, but I get a = 6.5.





    3.) It takes 1/200 Joules to stretch a spring from its natural lenth to another y centimeters. It takes 9/200 Joules to stretch the same spring from its natural length to another y + 4 centimeters. Find the value of y. What is the value of the spring constant?

    y = 2 and the spring constant is 25. I don't know how to start.






    4.) The region bounded by y = 0, x = 0, and y= (3-x) / π is rotated around the y-axis to form a solid of revolution. use the disc method to find the volume.

    The answer is 9. I was able to do it with the shell method, but when I used the
    disc method, I got some ridiculous answer that included π ^2.




    5.) Suppose it is always true that 2 ≤ f' (x) ≤ 4. What is the smallest possible value of f (7) - f (2)?
    The answer is 10.
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    Re: Solids of revolution, spring constants, and limits

    That's because you interpret the limit incorrect (and I admit it's not stated very clear!), if the answer is 4 then the limit should be:
    \lim_{x\to 10} \frac{\ln[(x-9)^8]}{2x-20}
    Do you note the difference?

    I wonder why you let t\to 10 in stead of x\to 10, probably a typo ... ?
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    Re: Solids of revolution, spring constants, and limits

    3.) It takes 1/200 Joules to stretch a spring from its natural lenth to another y centimeters. It takes 9/200 Joules to stretch the same spring from its natural length to another y + 4 centimeters. Find the value of y. What is the value of the spring constant?

    y = 2 and the spring constant is 25. I don't know how to start.

    4.) The region bounded by y = 0, x = 0, and y= (3-x) / π is rotated around the y-axis to form a solid of revolution. use the disc method to find the volume.

    The answer is 9. I was able to do it with the shell method, but when I used the
    disc method, I got some ridiculous answer that included π ^2.

    5.) Suppose it is always true that 2 ≤ f' (x) ≤ 4. What is the smallest possible value of f (7) - f (2)?
    The answer is 10.
    (3) F = kx

    \int_0^y kx \, dx = \frac{1}{200}

    \int_0^{y+.04} kx \, dx = \frac{9}{200}

    evaluate both definite integrals using the FTC ... this will give you two equations in k and y which you should be able to solve. Also, y = 2 cm , k = 25 N/m.

    --------------------------------------------------------------------------

    (4) y = \frac{3-x}{\pi}

    x = 3 - \pi y

    V = \pi \int_0^{3/\pi} (3 - \pi y)^2 \, dy

    --------------------------------------------------------------------------

    (5) 2 < \frac{f(7)-f(2)}{7-2} < 4 ... why?
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    Re: Solids of revolution, spring constants, and limits

    Quote Originally Posted by trevor22 View Post
    These are some problems that I'm having trouble with from a review packet. I know the answers, but I don't know how to do them. I would greatly appreciate any help I could get, even if you only help on one or two. I included my work for some of them so you can tell me what I'm doing wrong.


    1.) Find lim t --> 0 of [ln(x-9)^8] / [2x - 20]
    I used L'hospital's rule a few times and got 0, but the answer is 4. I see 8/2 in the last part, but I thought the ln in the numerator approaches 0...?








    2.) The region y = 2x - x^2 and y = 0 is rotated around the x-axis. Find the value of a if the resulting volume is 44π / 3.

    The answer is 3, but I get a = 6.5.





    3.) It takes 1/200 Joules to stretch a spring from its natural lenth to another y centimeters. It takes 9/200 Joules to stretch the same spring from its natural length to another y + 4 centimeters. Find the value of y. What is the value of the spring constant?

    y = 2 and the spring constant is 25. I don't know how to start.






    4.) The region bounded by y = 0, x = 0, and y= (3-x) / π is rotated around the y-axis to form a solid of revolution. use the disc method to find the volume.

    The answer is 9. I was able to do it with the shell method, but when I used the
    disc method, I got some ridiculous answer that included π ^2.




    5.) Suppose it is always true that 2 ≤ f' (x) ≤ 4. What is the smallest possible value of f (7) - f (2)?
    The answer is 10.
    For 2. what is "a" supposed to represent?

    For 4. note that the region you are rotating is a triangle, which means that the region will be a right-angle cone 3 units in radius, \displaystyle \begin{align*} \frac{3}{\pi} \end{align*} units in height.

    So the volume will be

    \displaystyle \begin{align*} V &= \frac{\pi r^2 h}{3} \\ &= \frac{\pi \cdot 3^2 \cdot \frac{3}{\pi}}{3} \\ &= 9 \textrm{ units}^2 \end{align*}

    Anyway, to use the discs method, visualise the area of that triangle being approximated using horizontal rectangular strips. They will have a length = x, and a width \displaystyle \begin{align*} = \Delta y \end{align*} (some small change in y.

    When you rotate these strips, you get cylinders (discs), that will have radius = x and height \displaystyle \begin{align*} = \Delta y \end{align*}

    So the volume of each disc is \displaystyle \begin{align*} \pi x^2 \Delta y = \pi \left(3 - \pi y\right)^2 \Delta y \end{align*}.

    So the volume of your region can be approximated by summing these cylinders.

    \displaystyle \begin{align*} V &\approx \sum{\pi \left(3 - \pi y\right)^2 \Delta y} \end{align*}

    As you increase the number of strips and make \displaystyle \begin{align*}  \Delta y \to 0 \end{align*}, the sum converges on an integral, and the approximation becomes exact. Note that your bounds will be y bounds...

    \displaystyle \begin{align*} V &= \int_0^{\frac{3}{\pi}}{\pi\left(3 - \pi y\right)^2\,dy} \\ &= -\int_3^{0}{u^2\,du} \textrm{ after making the substitution }u = 3 - \pi y \implies du = -\pi\,dy \\ &= \int_0^3{u^2\,du} \\ &= \left[\frac{u^3}{3}\right]_0^3 \\ &= \frac{3^3}{3} - \frac{0^3}{3} \\ &= 9 \end{align*}

    as required.
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    Re: Solids of revolution, spring constants, and limits

    Thanks. I was able to get all of them now except for 3. I wrote it wrong up there. It's supposed to say.

    2. The region between y = 2x - x^2 and y = 0 is rotated around the line x =a. Find the value of a if the resulting volume is 44π / 3.

    The answer is 3, but I get a = 6.5.

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