A right-angled triangle with sides 3 cm, 4 cm, and 5 cm is rotated about the side of 3 cm to form a cone. The volume of the cone so formed is?

1. A

   12

2. B

   20

3. C

   16

4. D

   15

Appeared in: SGPGI Nursing Officer-2022

Explanation

• When a right-angled triangle is rotated about one of its legs, that leg becomes the height (h) of the resulting cone, and the other leg becomes the radius (r).
• In this case, the triangle is rotated about the 3 cm side, so the height (h) is 3 cm and the radius (r) is the other leg, 4 cm.
• The volume of a cone is calculated using the formula V = (1/3) × π × r² × h.
• Substituting the values: V = (1/3) × π × (4)² × 3 = (1/3) × π × 16 × 3 = 16π cm³.
• The options provided are numerical values, implying the question asks for the coefficient of π. Therefore, the answer is 16.

Why Other Options Were Wrong

• Option A: This value is incorrect because it assumes the rotation happened around the 4 cm side, not the 3 cm side.
• Option B: This value does not correspond to the volume of the cone formed by rotating the 3-4-5 triangle around either of its legs.
• Option D: This value does not correspond to the volume of the cone formed by rotating the 3-4-5 triangle around either of its legs.

Related Visual

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Calculating the volume of a cone formed by rotating a right-angled triangle as background academic context rather than a clinical decision trigger.
• This is a general mathematics question testing knowledge of solid geometry.
• This type of question is included in nursing entrance exams to assess general aptitude and problem-solving skills, which are essential for a nursing career.
• While not directly related to patient care, the logical reasoning required is a foundational skill for nurses.

How to Approach the Question

• First, identify the shape being formed (a cone) and the 2D shape being rotated (a right-angled triangle).
• Determine the parameters of the cone from the rotation. The axis of rotation is the cone's height (h), and the other leg of the right triangle is the cone's radius (r).
• Recall the formula for the volume of a cone: V = (1/3) × π × r² × h.
• Substitute the identified values for 'r' and 'h' into the formula.
• Calculate the final volume. Check if the answer options include π or are just numerical values, and select the matching option.

Concept Tested & Keywords

• Concept Tested: Calculating the volume of a cone formed by rotating a right-angled triangle.
• Stem keywords: right-angled triangle, sides 3 cm, 4 cm, and 5 cm, rotated, side of 3 cm, cone, volume
• Lead-in keywords: The volume of the cone so formed is?
• Negative lead-in flag: false

Question ID

Q7vVJl9EUyuFzyc0oUKhDP

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