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.