AIIMS BHOPAL NO- 2018( Shift-2nd)
Non Nursing Subjects
Hard
The sum of the surface areas of two spheres is 100 cm² and the ratio of their volumes is 64:27. What is the sum of the radii of the spheres?
Appeared in: AIIMS BHOPAL NO- 2018( Shift-2nd)
Explanation
- The ratio of the volumes of two spheres (V₁/V₂ = r₁³/r₂³) is used to find the ratio of their radii (r₁/r₂). Given V₁/V₂ = 64/27, the ratio of radii r₁/r₂ is 4/3.
- The radii can be expressed as r₁ = 4x and r₂ = 3x.
- The formula for the sum of surface areas (4πr₁² + 4πr₂² = 100π) is used to create an equation: (4x)² + (3x)² = 25.
- Solving the equation gives x = 1.
- The sum of the radii is r₁ + r₂ = 4x + 3x = 7x. Substituting x=1 gives a sum of 7 cm.
Why Other Options Were Wrong
- Option B: This sum is too small and does not fit the mathematical relationship derived from the given volume and surface area values.
- Option C: This would imply a different ratio or size of the spheres. For example, if the radii were 3 cm and 3 cm, the sum would be 6 cm, but the volume ratio would be 1:1, not 64:27.
- Option D: This would imply larger radii. For example, if the radii were 4 cm and 4 cm, the sum would be 8 cm, but this contradicts the given 64:27 volume ratio.
Related Visual
Clinical Relevance
- Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving geometric problems involving spheres using formulas for volume and surface area as background academic context rather than a clinical decision trigger.
- This question assesses general aptitude and mathematical reasoning skills, which are essential for problem-solving in any field.
- While not directly clinical, the ability to work with formulas and perform multi-step calculations is a foundational skill for tasks like drug dosage calculations and interpreting quantitative data.
- What if the ratio of volumes was 8:1? Then r₁/r₂ = 2/1. Let r₁=2x, r₂=x. The surface area equation becomes (2x)² + x² = 25, so 5x²=25, x²=5, x=√5. The sum of radii would be 3x = 3√5 cm.
How to Approach the Question
- First, identify the key information provided: the sum of the surface areas (100π cm²) and the ratio of the volumes (64:27).
- Recall the formulas for a sphere's volume (V = 4/3πr³) and surface area (A = 4πr²).
- Use the volume ratio to determine the ratio of the radii. Since volume is proportional to the cube of the radius (V ∝ r³), the ratio of the radii will be the cube root of the ratio of the volumes.
- Express the two radii in terms of a single variable (e.g., r₁ = 4x, r₂ = 3x) based on the ratio you found.
- Substitute these expressions into the equation for the sum of the surface areas.
- Solve this equation to find the value of the variable 'x'.
Concept Tested & Keywords
- Concept Tested: Solving geometric problems involving spheres using formulas for volume and surface area.
- Stem keywords: surface areas, spheres, ratio of volumes, sum of radii
- Lead-in keywords: What is
- Negative lead-in flag: false
Question ID
QmSDriY8_xhH7lyAGcYbQk
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Attempt every question from this paper in a timed mock, then review the full solution for each one.
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