A solid is formed by mounting a right circular cone of height 12 cm and diameter 10 cm on a hemisphere with the same diameter 10 cm. What is the volume of the combined solid (in cubic centimetres)?

Introduction / Context:
This problem combines volumes of two standard solids: a right circular cone and a hemisphere. The cone is mounted on the flat face of the hemisphere, and both have the same radius because they share a common circular base. The question tests your ability to compute individual volumes and then add them to find the total volume of the composite solid.

Given Data / Assumptions:

• Diameter of both cone and hemisphere = 10 cm, so radius r = 5 cm.
• Height of the cone h = 12 cm.
• The hemisphere has radius 5 cm.
• We need the total volume of the solid formed by cone + hemisphere.
• Take π ≈ 3.14 (or equivalent approximation) to match the given numerical options.

Concept / Approach:
We use the standard formulas:

• Volume of a right circular cone: V_cone = (1/3) * π * r^2 * h.
• Volume of a hemisphere: V_hemi = (2/3) * π * r^3.

Since both shapes have the same radius, we compute each volume separately with r = 5 cm, then add them to get the total volume of the combined solid.

Step-by-Step Solution:
Step 1: Radius r = 10 / 2 = 5 cm. Step 2: Compute the volume of the cone: V_cone = (1/3) * π * r^2 * h = (1/3) * π * 5^2 * 12. Step 3: Evaluate r^2 = 25, so V_cone = (1/3) * π * 25 * 12 = (1/3) * 300π = 100π cubic centimetres. Step 4: Compute the volume of the hemisphere: V_hemi = (2/3) * π * r^3 = (2/3) * π * 5^3. Step 5: Evaluate r^3 = 125, so V_hemi = (2/3) * π * 125 = (250/3)π cubic centimetres. Step 6: Total volume V_total = V_cone + V_hemi = 100π + (250/3)π. Step 7: Combine terms: write 100π as (300/3)π, so V_total = (300/3)π + (250/3)π = (550/3)π. Step 8: Using π ≈ 3.1416, V_total ≈ (550/3) * 3.1416 ≈ 183.33 * 3.1416 ≈ 575.9 cubic centimetres. Step 9: Rounding appropriately and matching the closest option gives approximately 576.19 cubic centimetres.

Verification / Alternative check:
You can compute each volume numerically with π ≈ 3.1416. For the cone, V_cone ≈ 100 * 3.1416 ≈ 314.16 cubic centimetres. For the hemisphere, V_hemi ≈ (250/3) * 3.1416 ≈ 83.33 * 3.1416 ≈ 261.8 cubic centimetres. Summing these gives about 314.16 + 261.8 ≈ 575.96 cubic centimetres, which is extremely close to 576.19 cubic centimetres when rounded according to calculator precision, validating the chosen option.

Why Other Options Are Wrong:
192.06 cubic centimetres and 288.1 cubic centimetres are far too small and do not even match either individual volume, let alone the sum. 384.13 cubic centimetres is also significantly less than the required total and is not equal to either the cone or hemisphere volumes alone. Only 576.19 cubic centimetres matches the sum of the correctly computed component volumes.

Common Pitfalls:
A frequent mistake is to use the formula for a full sphere instead of a hemisphere, forgetting the factor 1/2 in volume. Another mistake is to use diameter instead of radius in the formulas, which would inflate the volumes by a factor of 4 or 8. Some learners also forget to convert the radius correctly when given diameter. Taking care to write radii, heights, and factors like 1/3 and 2/3 clearly avoids these issues.

Final Answer:
The volume of the combined solid is approximately 576.19 cubic centimetres.