A solid is in the shape of a cone standing on a hemisphere, with both their…

Concept: A composite solid is made of parts that occupy separate regions of space, so its volume is the sum of the volumes of those parts. Two standard formulas are needed here: a right circular cone of radius r and height h has volume (1/3)π r2 h, and a hemisphere of radius r has volume (2/3)π r3.

Application: Substitute the given measurements one part at a time.

1. Read off the dimensions: the cone and the hemisphere share the same radius, r = 1 cm, and the height of the cone equals its radius, so h = r = 1 cm.

2. Volume of the cone: (1/3)π r2 h = (1/3)π × (1)2 × 1 = π/3 cm3.

3. Volume of the hemisphere: (2/3)π r3 = (2/3)π × (1)3 = 2π/3 cm3.

4. Add the two parts: π/3 + 2π/3 = 3π/3 = π cm3.

Cross-check: Because the height of the cone equals the radius here, (1/3)π r2 h becomes (1/3)π r3, so the total is (1/3)π r3 + (2/3)π r3 = π r3; for this cone-on-a-hemisphere shape the volume is simply π r3. Putting r = 1 cm back in gives π cm3, which matches the step-by-step total. As a bound, the whole solid fits inside a cylinder of radius 1 cm and height 2 cm (1 cm of cone plus the 1 cm radius of the hemisphere), whose volume is 2π cm3, so the total must be smaller than 2π cm3.