# Sphere inside of a Sphere

I have a very interesting question.

I am given the fact that people like cookies the most when the ratio of cookie dough to chocolate is 1:1. The cookie is first placed on a sheet and has a diameter of one. The cookie is then covered in a layer of chocolate that's volume most match that of the cookie dough. It is a sphere inside of another sphere, with them having matching volumes.

• Where is the sphere? Do you mean a circle? Also, what do you want the fractional form of? Dec 9, 2019 at 23:27
• This sounds like a challenge for Claire Saffitz.
– TRiG
Jan 6, 2020 at 17:41

If the inner sphere of dough has radius $$r$$ and when the outer shell of choclate has radius $$R$$ ... and we require the volume of dough & chocolate to be equal then $$\begin{eqnarray*} 2 \frac{4 \pi r^3}{3} = \frac{4 \pi R^3}{3}. \end{eqnarray*}$$ So we require the ratio $$R/r = \sqrt{2}=1.2599 \cdots$$. So a good approximation would be $$\frac{5}{4}$$.

In other words if your ball of dough is $$4$$ units then when it is covered in chocolate make the radius increase to $$5$$ units.

Note that we rounded $$1.2599$$ down to $$1.25$$ so there will be slightly less chocolate ... so I would advise you to put an extra smidge of chocolate on in order to remedy this ... you can't put too much chocolate on your cookies!

Consider cookie as $$3D$$ object. So if corresponding $$1D$$ size of cookie + chocolate is $$a > 1$$ where $$1D$$ size of cookie is $$1$$, then $$3D$$ size of cookie with chocolate will be $$a^3$$.

Now, you have

$$\frac{cookie + chocolate}{cookie} = \frac{a^3}{1} = 2$$

Or

$$a^3 = 2$$

Therefore chocolate addition to diameter must be $$\sqrt{2} - 1$$