A right circular cylinder circumscribes a hemisphere so that they have the same circular base. Find the ratio of the volume of the cylinder to the volume of the hemisphere.

Introduction / Context:
This question compares the volumes of two common solids in mensuration: a right circular cylinder and a hemisphere. The cylinder is said to circumscribe the hemisphere with a common base, which means the hemisphere fits exactly inside the cylinder with the same radius. Understanding such relationships helps in visualizing three dimensional shapes and in quickly working with volume ratios in aptitude exams.

Given Data / Assumptions:

• The hemisphere and the cylinder share the same circular base.
• Let the common radius be r.
• The height of the hemisphere is r.
• The cylinder circumscribes the hemisphere, so its height equals the height of the hemisphere.
• We want the ratio: volume of cylinder : volume of hemisphere.

Concept / Approach:
We recall the volume formulas:
Volume of cylinder Vc = π * r^2 * h Volume of hemisphere Vh = (2 / 3) * π * r^3 Because the cylinder circumscribes the hemisphere with the same base, its height h equals r. Substituting h = r allows a direct comparison of the two volumes in terms of r and π, both of which will cancel in the ratio.

Step-by-Step Solution:
Step 1: Take height of cylinder h = r. Step 2: Volume of cylinder Vc = π * r^2 * r = π * r^3. Step 3: Volume of hemisphere Vh = (2 / 3) * π * r^3. Step 4: Form the ratio Vc : Vh = π * r^3 : (2 / 3) * π * r^3. Step 5: Cancel π * r^3 from numerator and denominator, giving 1 : (2 / 3). Step 6: Convert 1 : (2 / 3) to a ratio of whole numbers by multiplying both terms by 3, giving 3 : 2.

Verification / Alternative check:
We can also insert a specific value, such as r = 1 unit. Then volume of cylinder is π * 1^3 = π, and volume of hemisphere is (2 / 3) * π. The ratio π : (2 / 3)π again simplifies to 3 : 2. This numeric check confirms that the algebraic ratio is correct and independent of the chosen radius, because the radius cancels out.

Why Other Options Are Wrong:
5 : 2 would mean the cylinder volume is much larger than it really is compared to the hemisphere, which does not follow from the standard formulas.
7 : 2 and 9 : 2 exaggerate the cylinder volume even more and have no basis in the geometry of the solids.
1 : 3 would suggest that the hemisphere is three times as large as the cylinder by volume, which is clearly impossible because the hemisphere fits inside the cylinder.

Common Pitfalls:
A common mistake is to confuse height with diameter and to take the cylinder height as 2r instead of r. That would correspond to a cylinder that encloses a sphere, not a hemisphere. Another error is forgetting the correct volume formula for a hemisphere and using 4 / 3 π r^3, which is the formula for a full sphere. Careful reading of the problem and recalling the correct formulas prevent such errors.

Final Answer:
The ratio of the volume of the cylinder to the volume of the hemisphere is 3 : 2.