Understanding the Sphere Splitting and Moulding Problem

This question involves understanding how the volume of a sphere is divided and then how these parts are reformed into different 3D shapes: a cone and a cylinder. The key principle here is that the volume of the material remains constant when it is moulded into a new shape. We need to use the formulas for the volume of a sphere, a cone, and a cylinder, along with the given conditions about the dimensions of the cone and cylinder.

Step-by-Step Solution: Calculating Volumes and Radii

Let's denote the original sphere's radius by $R$.

The volume of the original sphere is given by the formula:

• Volume of Sphere ($V$) $= \frac{4}{3}\pi R^3$

The sphere is split into two parts in the ratio $1 : 3$. The total ratio parts are $1 + 3 = 4$.

Volume of the Smaller Part

The smaller part corresponds to the ratio 1 out of 4 total parts.

• Volume of Smaller Part ($V_s$) $= \frac{1}{4} \times V = \frac{1}{4} \times \frac{4}{3}\pi R^3 = \frac{1}{3}\pi R^3$

This smaller part is moulded into a cylinder.

Volume of the Larger Part

The larger part corresponds to the ratio 3 out of 4 total parts.

• Volume of Larger Part ($V_l$) $= \frac{3}{4} \times V = \frac{3}{4} \times \frac{4}{3}\pi R^3 = \pi R^3$

This larger part is moulded into a cone.

Analysing the Cone and Cylinder Dimensions

The Cone

The larger part ($V_l$) is moulded into a cone. Let the radius of the cone's base be $r_c$ and its height be $h_c$.

Given condition for the cone: height is equal to the radius of its base, so $h_c = r_c$.

The volume of the cone is given by:

• Volume of Cone ($V_{cone}$) $= \frac{1}{3}\pi r_c^2 h_c$

Substitute $h_c = r_c$ into the volume formula:

• $V_{cone} = \frac{1}{3}\pi r_c^2 (r_c) = \frac{1}{3}\pi r_c^3$

Since the larger part of the sphere is moulded into this cone, their volumes are equal:

• $V_{cone} = V_l$
• $\frac{1}{3}\pi r_c^3 = \pi R^3$

Now, we can solve for $r_c$ in terms of $R$:

• Divide both sides by $\pi$: $\frac{1}{3} r_c^3 = R^3$
• Multiply both sides by 3: $r_c^3 = 3R^3$
• Take the cube root of both sides: $r_c = \sqrt[3]{3R^3} = \sqrt[3]{3} \times \sqrt[3]{R^3} = \sqrt[3]{3} R$

The Cylinder

The smaller part ($V_s$) is moulded into a cylinder. Let the radius of the cylinder's base be $r_{cy}$ and its height be $h_{cy}$.

Given condition for the cylinder: height is equal to the radius of its base, so $h_{cy} = r_{cy}$.

The volume of the cylinder is given by:

• Volume of Cylinder ($V_{cyl}$) $= \pi r_{cy}^2 h_{cy}$

Substitute $h_{cy} = r_{cy}$ into the volume formula:

• $V_{cyl} = \pi r_{cy}^2 (r_{cy}) = \pi r_{cy}^3$

Since the smaller part of the sphere is moulded into this cylinder, their volumes are equal:

• $V_{cyl} = V_s$
• $\pi r_{cy}^3 = \frac{1}{3}\pi R^3$

Now, we can solve for $r_{cy}$ (and thus $h_{cy}$) in terms of $R$:

• Divide both sides by $\pi$: $r_{cy}^3 = \frac{1}{3}R^3$
• Take the cube root of both sides: $r_{cy} = \sqrt[3]{\frac{1}{3}R^3} = \sqrt[3]{\frac{1}{3}} \times \sqrt[3]{R^3} = \frac{1}{\sqrt[3]{3}} R$

Since $h_{cy} = r_{cy}$, the height of the cylinder is $h_{cy} = \frac{1}{\sqrt[3]{3}} R$.

Finding the Required Ratio

The question asks for the ratio of the radius of the cone ($r_c$) to the height of the cylinder ($h_{cy}$).

Ratio $= \frac{r_c}{h_{cy}}$

Substitute the expressions we found for $r_c$ and $h_{cy}$ in terms of $R$:

Ratio $= \frac{\sqrt[3]{3} R}{\frac{1}{\sqrt[3]{3}} R}$

We can cancel $R$ from the numerator and denominator (assuming $R \neq 0$, which must be true for a sphere to exist).

Ratio $= \frac{\sqrt[3]{3}}{\frac{1}{\sqrt[3]{3}}} = \sqrt[3]{3} \times \sqrt[3]{3}$

Using the property of cube roots $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}$:

Ratio $= \sqrt[3]{3 \times 3} = \sqrt[3]{9}$

So, the ratio of the radius of the cone to the height of the cylinder is $\sqrt[3]{9} : 1$.

Summary of Volumes and Dimensions

Ratio Calculation Steps

Ratio of $r_c$ to $h_{cy}$:

• $\frac{r_c}{h_{cy}} = \frac{\sqrt[3]{3} R}{\frac{1}{\sqrt[3]{3}} R}$
• $= \frac{\sqrt[3]{3}}{\frac{1}{\sqrt[3]{3}}}$
• $= \sqrt[3]{3} \times \sqrt[3]{3}$
• $= \sqrt[3]{3 \times 3}$
• $= \sqrt[3]{9}$

The ratio is $\sqrt[3]{9} : 1$.

Revision Table: Geometric Volumes and Formulas

Additional Information: Conservation of Volume

A fundamental concept used in this problem is the conservation of volume. When a solid object is melted, reshaped, or moulded into a new form without adding or removing material, its total volume remains the same. In this case, the volumes of the smaller and larger parts of the sphere are conserved when they are moulded into the cylinder and cone, respectively. This principle allows us to equate the volume of the sphere parts to the volumes of the resulting cylinder and cone, which is crucial for solving the problem.