Mensuration: Cylinder, Cone and Sphere

A. The Fundamental Law of Conservation of Volume:

When a metallic solid is melted and recast into another shape (or into multiple smaller identical solids):

B. Number of Spheres Formed Formula:

Example: If a metallic sphere of radius $R = 6\text{ cm}$ is melted to form small spherical balls of radius $r = 1\text{ cm}$:
$$n = \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3} = \left(\frac{R}{r}\right)^3 = \left(\frac{6}{1}\right)^3 = 216 \text{ balls}.$$

Problem:

A wooden toy is in the shape of a cone mounted on a hemisphere with identical base radius $r = 3.5\text{ cm}$. The total height of the toy is $15.5\text{ cm}$. Calculate (taking $\pi = 22/7$):

1. The total surface area of the toy.
2. The volume of the toy.

Solution:

Radius of hemisphere and cone $r = 3.5\text{ cm} = 7/2\text{ cm}$.
Height of hemisphere = radius = $3.5\text{ cm}$.
Height of conical part $h = \text{Total height} - \text{radius} = 15.5 - 3.5 = \mathbf{12\text{ cm}}$.
Slant height of cone $l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + (12)^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = \mathbf{12.5\text{ cm}}$.

1. Total Surface Area of Toy:
The circular base is enclosed inside and is NOT exposed! TSA = CSA of cone + CSA of hemisphere:
$$\text{TSA} = \pi r l + 2\pi r^2 = \pi r (l + 2r) = \frac{22}{7} \times \frac{7}{2} \times [12.5 + 2(3.5)] = 11 \times [12.5 + 7] = 11 \times 19.5 = \mathbf{214.5\text{ cm}^2}.$$

2. Volume of Toy:
$$\text{Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3 = \frac{1}{3}\pi r^2 (h + 2r)$$ $$\text{Volume} = \frac{1}{3} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times [12 + 7] = \frac{77}{6} \times 19 = \frac{1463}{6} = \mathbf{243.83\text{ cm}^3}.$$

A. Archimedes Principle in Mensuration:

Problem: A cylindrical vessel of diameter $12\text{ cm}$ is partly filled with water. A solid spherical iron ball of radius $3\text{ cm}$ is completely immersed in the water. Find the rise in the water level in the cylinder.

Solution:
Radius of cylinder $R = 12 / 2 = 6\text{ cm}$.
Volume of spherical ball immersed $V_{\text{sphere}} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi(27) = 36\pi\text{ cm}^3$.
Let the rise in water level be $h\text{ cm}$.
Volume of displaced cylindrical water $= \pi R^2 h = \pi (6)^2 h = 36\pi h$.
Equating volumes:
$$36\pi h = 36\pi \implies \mathbf{h = 1\text{ cm}}.$$
The water level rises by exactly $1\text{ cm}$.

Rigorous Step-by-Step Solutions for Top-Band Scores:

Below is a curated compendium of standard ICSE board-level examination questions designed to test mathematical derivation, algebraic accuracy, unit precision, and rigorous geometric justifications.

How ICSE Examiners Award Marks in this Topic:

In the official CISCE evaluation rubrics, marks are systematically distributed across three distinct cognitive dimensions:

• Method Marks (M): Awarded for writing the correct formula, establishing the proper geometric theorem, setting up the correct equation, or constructing the proper table columns. Even if a careless calculation error occurs later, method marks are fully preserved!
• Accuracy Marks (A): Awarded for correct intermediate arithmetic steps, accurate factorization, algebraic simplification, and correct radical reduction.
• Final Statement & Unit Marks (B/A): Awarded for stating the final numerical answer with correct units, proper rounding (e.g. two decimal places), and answering all sub-parts explicitly.

Top 5 Practical Recommendations to Maximize Scores:

1. Never skip writing the standard formula: Always write out the formula in algebraic terms before substituting numerical values.
2. Keep rough calculations neatly organized: Draw a dedicated 2-inch rough margin on the right side of your answer sheet for long divisions and square root extractions.
3. Check boundary conditions: Ensure your final solutions belong strictly to the specified domain (e.g., natural numbers, positive lengths, valid quadrants).
4. State geometric reasons in parentheses: In geometry proofs, every equality must be accompanied by its supporting theorem name (e.g., '[angles in the same segment are equal]').
5. Proofread before moving to the next question: Spend 30 seconds verifying signs, basic arithmetic, and decimal point placements.

Essential Concept & Formula Summary:

To ensure total retention and swift revision under timed examination conditions, internalize the following consolidated principles:

• Axiomatic Rigor: Always verify that intermediate algebraic transformations satisfy underlying domain constraints. For square roots, ensure expressions under the radical are non-negative; for fractional expressions, ensure denominators are strictly non-zero.
• Standard Mathematical Notations: Use formal set-builder notations for solution sets, standard vector/matrix bracket conventions, and accurate geometric angle labels (e.g., using three-letter vertex angle descriptions like $\angle ABC$ rather than ambiguous single-letter labels).
• Systematic Verification: In algebraic solving (equations, factorisation, matrix equations, and proportions), always substitute final numerical solutions back into the original question to confirm that both Left-Hand and Right-Hand Sides evaluate to exact numerical equality.
• Graph Paper Guidelines: When working on coordinate geometry, reflections, histograms, and ogives, always indicate the origin clearly, write the axis names, write the chosen scale at the top right, and mark plotted points with neat small circles or crosses.