Mold math challenge

[houseofwool]

I am looking to create a soap using a cylinder mold. It needs to weight ~30 gm cured and will be cut at 3/8". I cannot figure out the diameter of the mold.

A brand new soap that I just cut is 2"x2"x1". It weighs 67 gm.

A fully cured soap is 2-3/8"x2-7/16"X1". It weighs 126 gm.



Here's where I am not sure.

I calculate the density of the fresh soap as 16.75 gm/in3.

The cured soap is 15.29, giving a loss of 8.7%.

This should mean that the new soap should ~32.5 gm when cut.

I cannot wrap my head around how to figure out the new diameter. Help???

 

[MorpheusPA]

>>I calculate the density of the fresh soap as 16.75 gm/in3.

>>This should mean that the new soap should ~32.5 gm when cut.

Therefore, the soap's volume should be 32.5/16.75 = 1.94 cubic inches.

Since you plan on cutting at 3/8" (0.375"), the cylindrical surface area of the soap must be 1.94/0.375 = 5.17 square inches.

The equation for the derivative of the cylindrical surface area is easy--that's a circle. Good old pi-r-squared.

So pi*r^2 = 5.17. Divide both sides by pi.

So r^2 = 1.65". Take the square root.

r = 1.28". Call it 1.3"

The internal diameter of the cylinder should be 2.6" (twice the radius).

Back-work that to check...so...

1.3*1.3*pi*0.375 = 1.99 cubic inches (a touch higher than required due to always being generous with the rounding).

1.99 cu in * 16.75 gm/cu in = 33.33 grams.

Post cure, assuming that 8% loss you quoted, 33.33 * 0.92 = 30.7 grams.

So yes, a cylinder of 1.3" radius (or 2.6" diameter) is extremely close.

 

[MorpheusPA]

I just realized there's a discrepancy in your numbers above.

>>A fully cured soap is 2-3/8"x2-7/16"X1". It weighs 126 gm

Cool. Your 2.375 x 2.438 x 1.00 inch soap has a volume of 5.78 cubic inches.

Which makes the soap density 126/5.78 = 21.8 g/cu in.

Whoops! This density is considerably higher than you estimated above. That's going to change the answer. I used a rectangle shape as you didn't specify, which will give the lowest density as it has the greatest volume for those specifications. An ellipse would make the density even higher.

 

[Cactuslily]

MorpheusPA, now you're just showing off
You are amazing, and such an asset to my learning. Thank you!
Please write a book, or booklet...I'd buy it!

 

[houseofwool]

Gah - typo! The cured soap is 3-3/8 x 2-7/16 x 1


I'm pretty sure the density is correct.

 

[MorpheusPA]

Gah - typo! The cured soap is 3-3/8 x 2-7/16 x 1


I'm pretty sure the density is correct.

That ends up being 15.31 g/cu in, which while it doesn't agree is at least in the same ballpark as the above numbers--and between them.

So the size I gave actually works, given that soap density will vary somewhat. Air whipped soap would be somewhat under your target of 30, very dense gelled soaps somewhat over that. But it'll be close.