Final answer:
The question involves calculating the surface areas of a cone, a hemisphere, and a cylinder to determine their ratio. The areas are found using geometric formulas and the given equality of their bases and heights.
Step-by-step explanation:
The question asks for the ratio of the whole surface areas between three geometric shapes: a cone, a hemisphere, and a cylinder.
All shapes stand on equal bases with radius R and have equal heights H. To find the ratios, we must calculate the surface area for each shape and compare them.
For a cone, the lateral surface area is πRl (where l is the slant height) and the base area is πR^2. For a hemisphere, the surface area (including the base) is 3πR^2. Lastly, for a cylinder, the surface area is 2πRH (lateral) plus 2πR^2 (top and bottom bases).
To make further deductions, we need to use the given relations, R and H are equal among the shapes, and consider the slant height l of the cone which can be found through the Pythagorean theorem (l = √(R^2 + H^2)).
From this information, we can calculate and compare the total surface areas to find the ratio sought in the question.