The radius of a sphere and the radius of a right circular cylinder are both equal to r. If their volumes are equal, then what is the ratio of the height of the cylinder to its radius?

Introduction / Context:
This mensuration question compares the volume of a sphere and the volume of a right circular cylinder when both share the same radius. It tests your ability to use standard volume formulas and relate them through algebra to find an unknown ratio, here the ratio of the cylinder's height to its radius.

Given Data / Assumptions:

• A sphere has radius r.
• A right circular cylinder also has radius r.
• The volume of the sphere and the volume of the cylinder are equal.
• We must find the ratio (height of cylinder) : (radius of cylinder).
• pi is treated as a constant that will cancel out in the ratio.

Concept / Approach:
The volume of a sphere of radius r is:
V_sphere = (4 / 3) * pi * r^3.
The volume of a cylinder of radius r and height h is:
V_cylinder = pi * r^2 * h.
Since the volumes are equal, we set V_sphere = V_cylinder and solve for h in terms of r. From this expression we can directly read the ratio h : r.

Step-by-Step Solution:
Step 1: Write the equality of volumes: (4 / 3) * pi * r^3 = pi * r^2 * h. Step 2: Cancel the common factor pi from both sides. Step 3: Cancel r^2 on both sides of the equation, leaving (4 / 3) * r = h. Step 4: So h = (4 / 3) * r. Step 5: The ratio of height to radius is h : r = (4 / 3) * r : r. Step 6: Divide both parts of the ratio by r: h : r = 4 / 3 : 1 = 4 : 3.

Verification / Alternative check:
To verify, you can choose a convenient value for r, for example r = 3 units. Then h = (4 / 3) * 3 = 4 units. Volume of sphere = (4 / 3) * pi * 3^3 = (4 / 3) * pi * 27 = 36 * pi. Volume of cylinder = pi * 3^2 * 4 = pi * 9 * 4 = 36 * pi. The volumes match, confirming that the ratio 4 : 3 is correct for all r.

Why Other Options Are Wrong:
3 : 1 and 2 : 1: These ratios give heights that are too large relative to the radius and do not satisfy the equality of volumes.
3 : 2: This would make h = 1.5 * r, which does not yield equal volumes when substituted into the formulas.
1 : 1: This implies a cylinder with height equal to radius, clearly not enough volume to match a sphere with the same radius.

Common Pitfalls:
A common mistake is to confuse the formulas for volume and surface area, or to forget to cancel out common factors like pi and r^2. Some students also mistakenly cross multiply incorrectly or simplify the ratio h : r without first solving for h. Working step by step and cancelling common factors early makes the algebra easier and less error prone.

Final Answer:
The ratio of the height of the cylinder to its radius is 4 : 3.