A solid metallic sphere of radius 12 cm is melted and recast into a cone having diameter of the base as 12 cm. What is the height of the cone?
192 cm
The problem asks us to find the height of a cone that is formed by melting and recasting a solid metallic sphere. A key principle in such problems is that the volume of the material remains constant during the transformation from one shape to another. Therefore, the volume of the original sphere is equal to the volume of the newly formed cone.
Let's first find the volume of the sphere.
The radius of the sphere is given as 12 cm.
The formula for the volume of a sphere is \(V_{sphere} = \frac{4}{3} \pi r^3\), where \(r\) is the radius of the sphere.
Given \(r = 12\) cm, the volume of the sphere is:
\[ V_{sphere} = \frac{4}{3} \pi (12 \text{ cm})^3 \] \[ V_{sphere} = \frac{4}{3} \pi (1728 \text{ cm}^3) \]Now, we can calculate the numerical value:
\[ V_{sphere} = 4 \pi \times \frac{1728}{3} \] \[ V_{sphere} = 4 \pi \times 576 \text{ cm}^3 \] \[ V_{sphere} = 2304 \pi \text{ cm}^3 \]Next, let's consider the cone. The metallic sphere is recast into a cone having a diameter of the base as 12 cm. We need to find the height of this cone.
The diameter of the base of the cone is 12 cm. The radius of the base of the cone, \(r_{cone}\), is half of the diameter.
\[ r_{cone} = \frac{\text{Diameter}}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]The formula for the volume of a cone is \(V_{cone} = \frac{1}{3} \pi r_{cone}^2 h\), where \(r_{cone}\) is the radius of the base and \(h\) is the height of the cone.
Given \(r_{cone} = 6\) cm, the volume of the cone is:
\[ V_{cone} = \frac{1}{3} \pi (6 \text{ cm})^2 h \] \[ V_{cone} = \frac{1}{3} \pi (36 \text{ cm}^2) h \] \[ V_{cone} = 12 \pi h \text{ cm}^3 \]Since the volume of the material is conserved when the sphere is melted and recast into the cone, the volume of the sphere is equal to the volume of the cone.
\[ V_{sphere} = V_{cone} \] \[ 2304 \pi \text{ cm}^3 = 12 \pi h \text{ cm}^3 \]To find the height \(h\), we can divide both sides of the equation by \(12 \pi\).
\[ h = \frac{2304 \pi \text{ cm}^3}{12 \pi \text{ cm}^2} \]The \(\pi\) terms cancel out, and the units become cm (\(cm^3 / cm^2 = cm\)).
\[ h = \frac{2304}{12} \text{ cm} \]Now, let's perform the division:
| Calculation | Result |
|---|---|
| $2304 \div 12$ | 192 |
\[ h = 192 \text{ cm} \]
So, the height of the cone is 192 cm.
Let's check the given options:
- Option 1: 258 cm
- Option 2: 192 cm
- Option 3: 166 cm
- Option 4: 224 cm
Our calculated height is 192 cm, which matches Option 2.
Understanding Melted and Recast Shapes
When a solid object is melted and recast into another shape, the total amount of material remains the same. This means the total volume of the material does not change. This principle is fundamental in solving problems involving transformations of shapes.
In this specific problem:
- The original shape is a solid metallic sphere.
- Its volume is calculated using the sphere volume formula and its given radius.
- The new shape is a cone.
- Its volume is calculated using the cone volume formula, its base radius (derived from the diameter), and its height (which is unknown).
- By equating the volume of the sphere to the volume of the cone, we form an equation that can be solved for the unknown height of the cone.
Sphere and Cone Volume Formulas
It is important to remember the volume formulas for basic 3D shapes:
- Sphere: \( V = \frac{4}{3} \pi r^3 \) where \(r\) is the radius.
- Cone: \( V = \frac{1}{3} \pi r^2 h \) where \(r\) is the radius of the base and \(h\) is the height.
- Cylinder: \( V = \pi r^2 h \) where \(r\) is the radius of the base and \(h\) is the height.
These formulas are essential for solving problems involving melting and recasting solids.
Step-by-Step Solution Recap
- Identify the initial shape (Sphere) and its dimensions (radius).
- Calculate the volume of the initial shape using the appropriate formula.
- Identify the final shape (Cone) and its known dimensions (base diameter, from which radius is found). Identify the unknown dimension (height).
- Write down the formula for the volume of the final shape, including the unknown variable.
- Equate the volume of the initial shape to the volume of the final shape based on the conservation of volume principle.
- Solve the resulting equation for the unknown dimension (height of the cone).
- Ensure units are consistent throughout the calculation.
Revision Table: Key Information
| Shape | Given/Derived Dimension | Volume Formula |
|---|---|---|
| Sphere | Radius (r) = 12 cm | $\frac{4}{3} \pi r^3$ |
| Cone | Diameter = 12 cm Radius (rcone) = 6 cm Height (h) = ? |
$\frac{1}{3} \pi r_{cone}^2 h$ |
Additional Information on Volume Conservation
The principle of conservation of volume in melting and recasting problems is based on the assumption that there is no loss of material during the process. This is a standard assumption in these types of mathematical problems unless stated otherwise. When a solid metal is melted, it turns into a liquid, but the amount of metal remains the same. When this liquid metal is poured into a mould to form a new shape, the volume occupied by the metal in the new shape is equal to the volume it occupied in the original shape.
This principle applies to transformations between any two or more shapes, as long as the total mass of the material is conserved, and the density remains constant (which is generally true for a specific material). For example, a sphere could be melted and recast into multiple smaller spheres, or into cylinders, cubes, or a combination of shapes. In all such cases, the sum of the volumes of the final shapes will equal the volume of the original shape.
Understanding volume formulas and the concept of volume conservation is crucial for solving mensuration problems involving shape transformations.
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