Answer:
To calculate the values of ( a ) and ( b ), we can compare the formulas for the curved surface area of a cone and a hemisphere, and the volume of the shape given.
The curved surface area of a cone is given by ( \pi rl ), where ( r ) is the radius of the base of the cone and ( l ) is the slant height.
The curved surface area of a hemisphere is half the surface area of a sphere, which is ( 2\pi r^2 ).
Given that the cone and hemisphere have the same curved surface area, we can equate the two formulas:
[ \pi rl = 2\pi r^2 ] [ rl = 2r ] [ l = 2 ]
The volume of the shape can be written as ( \frac{1}{3} \pi r^2 h + \frac{2}{3} \pi r^3 ), which simplifies to ( \frac{1}{3} \pi r^2 (2r) + \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 (1 + \sqrt{3}) ).
Comparing this with the given form:
[ a = 2 ] [ b + \sqrt{\sqrt{3}} = 1 + \sqrt{3} ] [ b = 1 ]
Therefore, the values of ( a ) and ( b ) are:
( a = 2 )
( b = 1 )
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