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Math Help - shape of tapered bottle.

  1. #1
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    shape of tapered bottle.

    I wonder if anyone can help me. I need to design a sticker to cover a jar which will be used to collect money for charity. The relevant part of the jar is 405mm in circumference at the top, and 390mm in circumference at a point 85mm lower. How can I plot the shape to use as my outline on my computer, so that when printed and cut-out and stuck on it will be straight all round at the top of the jar?
    Hope I've made it all clear.
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  2. #2
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    Lexington, MA (USA)
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    Hello, bigginge!

    \text{Design a label to cover a jar used to collect money for charity.}
    \text{The jar is 405 mm in circumference at the top, and 390 mm}
    \text{at the bottom at a point 85 mm lower.}

    \text{How can I plot the shape to use as my outline on my computer,}
    \text{so that when printed and cut out and stuck on,}
    \text{it will be straight all around at the top and bottom of the jar?}

    We want a sector of an annulus (ring).

    Code:
                  * * *..
              *     |:::::* 405
            *       |:::::::*
           *        *:: 390 :*
                *   |   *:::::
          *    *    |@   *::::*
          *    *    *----*----*
          *    *-      r * 85 *
                *       *
           *        *        *
            *               *
              *           *
                  * * *

    The radius of the smaller circle is \,r.
    The sector has central angle \,\theta.
    The arc length on the smaller circle is 390 mm.

    The radius of the larger circle is \,r+85.
    The arc length on the lartger circle is 405 mm.


    Arc length formula: . s \:=\:r\theta . with \,\theta in radians.

    Smaller circle: . r\theta \:=\:390 .[1]

    Larger circle: . (r+85)\theta \:=\:405 \quad\Rightarrow\quad r\theta + 85\theta \:=\:405 .[2]

    Substitute [1] into [2]: . 390 + 85\theta \:=\:405

    And we have: . \theta \:=\:\frac{3}{17}\text{ radians}

    Substitute into [1]: . r(\frac{3}{17}) \:=\:390 \quad\Rightarrow\quad r \,=\, 2210


    Construct two concentric circles with radii 2210 mm and 2295 mm.
    Cut out a sector with central angle of \frac{3}{17} radians or about 10.1 degrees.

    The label is the curved region between the two circles.

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  3. #3
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    Joined
    Dec 2010
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    Thank you! It's been about 45 years since I got my maths O level and I didn't understand the first part, but I did the summary in Adobe Illustrator and it looks just right.
    Thanks again.
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