Sphere or Cylinder?

📌 CONCEPT: The problem requires us to compare the volumes of two similar right circular cylinders.

📐 RULE / FORMULA: The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.

💡 WORKED EXAMPLE: Let's consider the two cylinders: Cylinder 1 (r = 4 cm, h = 8 cm) and Cylinder 2 (r = 8 cm, h = 2 cm). Their volumes are V1 = π(4)^2(8) and V2 = π(8)^2(2). Since the radii are in the ratio 4:8 = 1:2, the heights should also be in the same ratio for the cylinders to be similar. Thus, the ratio of their volumes will be the same as the ratio of their radii squared. V2/V1 = (8/4)^2 = 4. Since 4 > 1, Cylinder 2 will have a larger volume.

⚠️ COMMON MISTAKE: Students often forget to consider the ratio of the radii and heights of the cylinders.