frac{13}{20}pi

Using the given volume and length of the cylinder, we can calculate it's radius r_1:

V_1 = \pi r_1^2h
25\pi = \pi r_1^2 \times 4
r_1^2 = 6.25
r_1 = 2.5

Since the radius r_2 of the base of the cone is twice the radius of the cylinder, r_2 = 2r_1 = 5 inches. By slicing the cone vertically and drawing the 12 inch line denoting it's length through the center of the figure, a right triangle with legs 5 and 12 is formed. Through the Pythagorean theorem, the hypothenuse h is:

h = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13

Using the formula for lateral surface area A_l of a cone:

A_l = \pi hr = 65\pi

Since the plated portion accounts for \frac{1}{10} of the total length of the cone, it's area A_p covers (\frac{1}{10})^2 = \frac{1}{100} of the total lateral surface area of the cone. Thus, the area of the plated portion needed in square inches is:

A_p = \frac{1}{100} \times A_l = \frac{1}{100} \times 65\pi = \frac{65}{100}\pi = \frac{13}{20}\pi

frac{9sqrt{2}}{4}

By visualizing a diagram of the current set of objects, we can identify the object formed by the four centers, as well as it's properties. Through this we find that the centers of the spheres form a regular tetrahedron, with side lengths 3 units. This regular tetrahedron can be visualized as a triangle-base pyramid with formula:

V = frac{1}{3}Ah

Where V is the volume, A is the area of the base, and h is the height. The area A of the base of the 3-unit per side equilateral triangle can be calculated with Heron's formula:

A = sqrt{s(s-a)(s-b)(s-c)}

Where a=b=c=3, s = frac{a+b+c}{2} = frac{9}{2},

A=sqrt{frac{9}{2} times frac{3}{2} times frac{3}{2} times frac{3}{2}} = frac{9sqrt{3}}{4}

Let point O be the foot of the perpendicular dropped from the apex of the triangular pyramid to its triangular base. The height of the pyramid can be calculated through the pythagorean theorem:

x^2 + h^2 = 3^2

Where x is the distance from a vertex of the triangle base to point O. Because an equilateral triangle has 60^circ internal angles,

xcos(30) = frac{3}{2}
frac{xsqrt{3}}{2} = frac{3}{2}
x = sqrt{3}

Thus,

(sqrt{3})^2 + h^2 = 3^2
h^2 = 9-3 = 6
h =sqrt{6}

V = frac{1}{3} times frac{9sqrt{3}}{4} times sqrt{6} = boxed{frac{9sqrt{2}}{4}}

4:3

We first set up a system of equations based on the given information. Since the surface area and radius of both objects are the same,

2pi r^2 + 2pi rh = 4\pi r^2
2pi rh = 2pi r^2
h = r

Thus, with h=r, we can solve for the ratio of the volume of the sphere to the volume of the cylinder:

frac{4}{3}\pi r^3 : \pi r^2h
frac{4}{3}r^3 : r^3
4:3