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Thread: 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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    Hello, bigginge!

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

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

    We want a sector of an annulus (ring).

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

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

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


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

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

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

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

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

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


    Construct two concentric circles with radii 2210 mm and 2295 mm.
    Cut out a sector with central angle of $\displaystyle \frac{3}{17}$ 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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    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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