(a) The largest possible hemisphere is drilled out from a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the cube. Find : (i) the volume of wood left in the block, (ii) the total surface area of the remaining solid. OR (b) A solid toy is in the form of a hemisphere surmounted by a right circular cone. Ratio of the radius of the cone to its slant height is $\frac{3}{5}$. If the volume of the toy is $240\pi \text{ cm}^3$, then find the total height of the toy.

Step 1: Identify dimensions of the cube and hemisphere

For the largest possible hemisphere to be drilled out from a cubical block, the diameter of the hemisphere must be equal to the side of the cube. Therefore, the radius of the hemisphere is half the side of the cube.

Step 2: Calculate the volume of wood left

The volume of wood left is the volume of the cube minus the volume of the hemisphere. We use the formulas for the volume of a cube ($a^3$) and a hemisphere ($\frac{2}{3}\pi r^3$) and substitute the given values.

Step 3: Compute the volume of wood left

Perform the calculations for the volume of the cube and the hemisphere. Then subtract the volume of the hemisphere from the volume of the cube to get the final volume of wood left.

Step 4: Calculate the total surface area of the remaining solid

The total surface area of the remaining solid includes the area of 5 faces of the cube, the area of the base of the cube not covered by the hemisphere, and the curved surface area of the hemisphere. This simplifies to $6a^2 + \pi r^2$.

Step 5: Compute the total surface area

Perform the calculations for the areas and sum them up to find the total surface area of the remaining solid.

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