Ball & Dome

From the question, it is obvious that the total surface area of a pink candy ball = $4\pi r^2$ .

For the blue dome, the total surface area = the base area + the spherical section area

Then the base area = $4\pi R^2$ .

Now in order to calculate the spherical section area, we need to find out the radius of the original full sphere, $S$ :

By using Pythagorean theorem, $S^2 = (S-h)^2 + R^2$ .

Thus, $2Sh = h^2 + R^2$ .

Hence, $S = \dfrac{h^2 + R^2}{2h}$ .

Now according to Archimedes' Hat-Box Theorem , any spherical section from spherical radius $R$ and of height $h$ will have its lateral surface area equal to the lateral surface area of a cylinder of radius $R$ and height $h$ :

That is, the spherical section area = $2\pi Sh = \pi (2h)\dfrac{h^2 + R^2}{2h} = \pi(h^2 + R^2)$

Therefore, the total surface area of a blue dome = $\pi R^2 + \pi(h^2 + R^2) = \pi (2R^2 + h^2$ ).

Setting up equation with $2r = R + h$ :

$4\pi r^2 = \pi (2r)^2 = \pi (R+h)^2 = \pi (2R^2 + h^2$ ).

$2Rh = R^2$

$R = 2h$ .

As a result, the ratio $\dfrac{R}{h} = \boxed{2}$ .